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A027466

Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j).

1, 7, 1, 49, 14, 1, 343, 147, 21, 1, 2401, 1372, 294, 28, 1, 16807, 12005, 3430, 490, 35, 1, 117649, 100842, 36015, 6860, 735, 42, 1, 823543, 823543, 352947, 84035, 12005, 1029, 49, 1, 5764801, 6588344, 3294172, 941192, 168070, 19208, 1372, 56, 1

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OFFSET

0,2

COMMENTS

T(i,j) is the number of i-permutations of 8 objects a,b,c,d,e,f,g,h, with repetition allowed, containing j a's. - Zerinvary Lajos, Dec 21 2007

Triangle of coefficients in the expansion of (7 + x)^n, where n is a nonnegative integer. - Zagros Lalo, Jul 21 2018

REFERENCES

Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 44, 48

LINKS

Harvey P. Dale, Table of n, a(n) for n = 0..5000

B. N. Cyvin et al., Isomer enumeration of unbranched catacondensed polygonal systems with pentagons and heptagons, Match, No. 34 (Oct 1996), pp. 109-121.

FORMULA

Cube of lower triangular normalized Binomial matrix.

Numerators of lower triangle of (a( i, j ))^3 where a( i, j ) = binomial(i-1, j-1)/2^(i-1) if j <= i, 0 otherwise.

T(0,0) = 1; T(n,k) = 7*T(n-1,k) + T(n-1,k-1) for k = 0..n; T(n,k)=0 for n or k < 0. - Zagros Lalo, Jul 21 2018

EXAMPLE

Triangle begins:

7, 1;

49, 14, 1;

343, 147, 21, 1;

2401, 1372, 294, 28, 1;

16807, 12005, 3430, 490, 35, 1;

117649, 100842, 36015, 6860, 735, 42, 1;

823543, 823543, 352947, 84035, 12005, 1029, 49, 1;

5764801, 6588344, 3294172, 941192, 168070, 19208, 1372, 56, 1;

MAPLE

for i from 0 to 8 do seq(binomial(i, j)*7^(i-j), j = 0 .. i) od; # Zerinvary Lajos, Dec 21 2007

MATHEMATICA

Flatten[Table[Binomial[i, j]7^(i-j), {i, 0, 10}, {j, 0, i}]] (* Harvey P. Dale, Dec 03 2012 *)

t[0, 0] = 1; t[n_, k_] := t[n, k] = If[n < 0 || k < 0, 0, 7 t[n - 1, k] + t[n - 1, k - 1]]; Table[t[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Zagros Lalo, Jul 21 2018 *)

Table[CoefficientList[ Expand[(7 + x)^n], x], {n, 0, 8}] // Flatten (* Zagros Lalo, Jul 22 2018 *)

PROG

(GAP) Flat(List([0..8], i->List([0..i], j->Binomial(i, j)*7^(i-j)))); # Muniru A Asiru, Jul 21 2018

CROSSREFS

Cf. A007318, A038207.

Cf. A317014

Sequence in context: A264617 A389085 A038267 * A218017 A075502 A052104

Adjacent sequences: A027463 A027464 A027465 * A027467 A027468 A027469

KEYWORD

nonn,tabl,easy

AUTHOR

Olivier Gérard

EXTENSIONS

Simpler definition from N. J. A. Sloane

STATUS

approved

URL: https://oeis.org/A027466

⇱ A027466 - OEIS