0,5

One of a family of Pascal-like arrays. A007318 is equivalent to the (1,0,1)-array. A008288 is equivalent to the (1,1,1)-array. Rows include A017077, A081593, A081594. Coefficients of the row polynomials in the Newton basis are given by A013614.

Vincenzo Librandi, Rows n = 0..100, flattened

T(n,k) = Sum_{j = 0..n-k} binomial(n-k,j)*binomial(k,j)*8^j.

Riordan array (1/(1 - x), x*(1 + 7*x)/(1 - x)).

Square array T(n, k) defined by T(n, 0) = T(0, k)=1, T(n, k) = T(n, k-1) + 7*T(n-1, k-1) + T(n-1, k).

Rows are the expansions of (1 + 7*x)^k/(1 - x)^(k+1).

T(n, k) = Hypergeometric2F1([-k, k-n], [1], 8). - Jean-François Alcover, May 24 2013

E.g.f. for the n-th subdiagonal, n = 0,1,2,..., equals exp(x)*P(n,x), where P(n,x) is the polynomial Sum_{k = 0..n} binomial(n,k)*(8*x)^k/k!. For example, the e.g.f. for the second subdiagonal is exp(x)*(1 + 16*x + 64*x^2/2) = 1 + 17*x + 97*x^2/2! + 241*x^3/3! + 449*x^4/4! + 721*x^5/5! + .... - Peter Bala, Mar 05 2017

Sum_{k=0..n} T(n, k) = A015519(n+1). - G. C. Greubel, May 26 2021

Rows begin

1, 1, 1, 1, 1, ... A000012;

1, 9, 17, 25, 33, ... A017077;

1, 17, 97, 241, 449, ... A081593;

1, 25, 241, 1161, 3297, ...

1, 33, 449, 3297, 14721, ...

Triangle begins:

1;

1, 1;

1, 9, 1;

1, 17, 17, 1;

1, 25, 97, 25, 1;

1, 33, 241, 241, 33, 1;

1, 41, 449, 1161, 449, 41, 1;

1, 49, 721, 3297, 3297, 721, 49, 1;

1, 57, 1057, 7161, 14721, 7161, 1057, 57, 1;

Table[ Hypergeometric2F1[-k, k-n, 1, 8], {n, 0, 10}, {k, 0, n}]//Flatten (* Jean-François Alcover, May 24 2013 *)

(Magma)

A081582:= func< n, k, q | (&+[Binomial(k, j)*Binomial(n-j, k)*q^j: j in [0..n-k]]) >;

[A081582(n, k, 7): k in [0..n], n in [0..12]]; // G. C. Greubel, May 26 2021

(SageMath) flatten([[hypergeometric([-k, k-n], [1], 8).simplify() for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 26 2021

Cf. Pascal (1,m,1) array: A123562 (m = -3), A098593 (m = -2), A000012 (m = -1), A007318 (m = 0), A008288 (m = 1), A081577 (m = 2), A081578 (m = 3), A081579 (m = 4), A081580 (m = 5), A081581 (m = 6), A143683 (m = 8).

Cf. A015519, A017077, A081593.

Sequence in context: A128060 A168625 A143681 * A174346 A144404 A014761

Adjacent sequences: A081579 A081580 A081581 * A081583 A081584 A081585

easy,nonn,tabl

Paul Barry, Mar 23 2003

approved