1,8

A370031(n) = Sum_{k=0..n-1} T(n,k), for n >= 1.

A355868(n) = Sum_{k=0..n-1} T(n,k) * 2^k, for n >= 1.

A370033(n) = Sum_{k=0..n-1} T(n,k) * 3^k, for n >= 1.

A370034(n) = Sum_{k=0..n-1} T(n,k) * 4^k, for n >= 1.

A370035(n) = Sum_{k=0..n-1} T(n,k) * 5^k, for n >= 1.

A370036(n) = Sum_{k=0..n-1} T(n,k) * 6^k, for n >= 1.

A370037(n) = Sum_{k=0..n-1} T(n,k) * 7^k, for n >= 1.

A370038(n) = Sum_{k=0..n-1} T(n,k) * 8^k, for n >= 1.

A370039(n) = Sum_{k=0..n-1} T(n,k) * 9^k, for n >= 1.

A370043(n) = Sum_{k=0..n-1} T(n,k) * 10^k, for n >= 1.

Paul D. Hanna, Table of n, a(n) for n = 1..2556

G.f. A(x,y) = Sum_{n>=1} T(n,k)*x^n*y^k satisfies the following formulas.

(1) Sum_{n=-oo..+oo} (x^n - y*A(x,y))^n = 1 - (y-2)*Sum_{n>=1} x^(n^2).

(2) Sum_{n=-oo..+oo} x^n * (x^n + y*A(x,y))^(n-1) = 1 - (y-2)*Sum_{n>=1} x^(n^2).

(3) Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n - y*A(x,y))^n = 0.

(4) Sum_{n=-oo..+oo} x^(n^2) / (1 - x^n*y*A(x,y))^n = 1 - (y-2)*Sum_{n>=1} x^(n^2).

(5) Sum_{n=-oo..+oo} x^(n^2) / (1 + x^n*y*A(x,y))^(n+1) = 1 - (y-2)*Sum_{n>=1} x^(n^2).

(6) Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 - x^n*y*A(x,y))^n = 0.

(7) A(x,y) = (1/y) * Integral Q(x) / Sum_{n=-oo..+oo} n * (x^n - y*A(x,y))^(n-1) dy, where Q(x) = Sum_{n>=1} x^(n^2).

(8) A(x,y=0) = (theta_3(x) - 1)/2 * Product_{n>=1} (1 - x^(4*n-2)) / (1 - x^(4*n)), which is the g.f. of column 0 (A370153) defined at y = 0.

G.f.: A(x,y) = x*(1) + x^2*(0 + y) + x^3*(-1 + y^2) + x^4*(1 - 3*y + y^3) + x^5*(1 + 6*y - 6*y^2 + y^4) + x^6*(-1 + 6*y + 19*y^2 - 10*y^3 + y^5) + x^7*(-2 - 18*y + 17*y^2 + 44*y^3 - 15*y^4 + y^6) + x^8*(1 - 4*y - 98*y^2 + 35*y^3 + 85*y^4 - 21*y^5 + y^7) + x^9*(4 + 36*y + 39*y^2 - 334*y^3 + 60*y^4 + 146*y^5 - 28*y^6 + y^8) + x^10*(-2 + 11*y + 291*y^2 + 311*y^3 - 879*y^4 + 91*y^5 + 231*y^6 - 36*y^7 + y^9) + ...

where

Sum_{n=-oo..+oo} (x^n - y*A(x,y))^n = 1 - (y-2)*Sum_{n>=1} x^(n^2).

TRIANGLE.

This triangle of coefficients T(n,k) of x^n*y^k in g.f. A(x,y) begins

1;

0, 1;

-1, 0, 1;

1, -3, 0, 1;

1, 6, -6, 0, 1;

-1, 6, 19, -10, 0, 1;

-2, -18, 17, 44, -15, 0, 1;

1, -4, -98, 35, 85, -21, 0, 1;

4, 36, 39, -334, 60, 146, -28, 0, 1;

-2, 11, 291, 311, -879, 91, 231, -36, 0, 1;

-5, -74, -264, 1310, 1286, -1960, 126, 344, -45, 0, 1;

3, -30, -627, -2547, 4248, 3935, -3892, 162, 489, -55, 0, 1;

6, 178, 773, -2626, -12982, 11138, 9989, -7092, 195, 670, -66, 0, 1;

-4, 40, 1525, 10094, -5842, -48126, 25138, 22258, -12093, 220, 891, -78, 0, 1;

...

(PARI) /* Generate A(x, y) by use of definition in name */

{T(n, k) = my(A=[0, 1]); for(i=1, n, A = concat(A, 0);

A[#A] = polcoeff( sum(m=-sqrtint(#A+1), #A, (x^m - y*Ser(A))^m ) - 1 + (y-2)*sum(m=1, sqrtint(#A+1), x^(m^2) ), #A-1)/y ); polcoeff(A[n+1], k, y)}

for(n=1, 15, for(k=0, n-1, print1(T(n, k), ", ")); print(""))

(PARI) /* Generate A(x, y) recursively using integration wrt y */

{T(n, k) = my(A = x +x*O(x^n), M=sqrtint(n+1), Q = sum(m=1, M, x^(m^2)) +x*O(x^n));

for(i=0, n, A = (1/y) * intformal( Q / sum(m=-M, n, m * (x^m - y*A)^(m-1)), y) +x*O(x^n));

polcoeff(polcoeff(A, n, x), k, y)}

for(n=1, 15, for(k=0, n-1, print1(T(n, k), ", ")); print(""))

Cf. A370030, A370031, A355868, A370033, A370034, A370035, A370036, A370037, A370038, A370039, A370043.

Cf. A370153 (column 0), A370154 (column 1), A370155 (column 2).

Cf. A370040 (dual triangle).

Sequence in context: A294212 A220691 A271023 * A143624 A126308 A370040

Adjacent sequences: A370038 A370039 A370040 * A370042 A370043 A370044

sign,tabl

Paul D. Hanna, Feb 10 2024

approved