A124835 -id:A124835 - cp's OEIS frontend

A124834 Triangle, read by rows, where the g.f. of column k, C_k(x), is equal to the product: C_k(x) = Product_{k=0..n} 1/(1 - binomial(n,k)*x).

1, 1, 1, 1, 2, 1, 1, 3, 4, 1, 1, 4, 11, 8, 1, 1, 5, 26, 42, 16, 1, 1, 6, 57, 184, 163, 32, 1, 1, 7, 120, 731, 1358, 638, 64, 1, 1, 8, 247, 2736, 10121, 10244, 2510, 128, 1, 1, 9, 502, 9844, 70436, 145475, 78320, 9908, 256, 1, 1, 10, 1013, 34448, 468735, 1911956, 2141835

Offset: 0

Paul D. Hanna, Nov 09 2006

			Column g.f.s begin:
C_0(x) = 1/(1-x);
C_1(x) = 1/((1-x)(1-x));
C_2(x) = 1/((1-x)(1-2x)(1-x));
C_3(x) = 1/((1-x)(1-3x)(1-3x)(1-x));
C_4(x) = 1/((1-x)(1-4x)(1-6x)(1-4x)(1-x)); ...
Triangle begins:
1;
1, 1;
1, 2, 1;
1, 3, 4, 1;
1, 4, 11, 8, 1;
1, 5, 26, 42, 16, 1;
1, 6, 57, 184, 163, 32, 1;
1, 7, 120, 731, 1358, 638, 64, 1;
1, 8, 247, 2736, 10121, 10244, 2510, 128, 1;
1, 9, 502, 9844, 70436, 145475, 78320, 9908, 256, 1;
1, 10, 1013, 34448, 468735, 1911956, 2141835, 604160, 39203, 512, 1; ...

Cf. A124835 (row sums), A124836 (central terms).

{T(n,k)=polcoeff(1/prod(j=0,k,1-binomial(k,j)*x +x*O(x^n)),n-k)}

T(n+1,n) = 2^n. T(n+2,n) = A032443(n) = Sum_{i=0..n} binomial(2*n,i).

A124836 Central terms of even-indexed rows in triangle A124834.

Original entry on oeis.org

1, 2, 11, 184, 10121, 1911956, 1277642344, 3076635199744, 27117951046505365, 883613507047099010632, 107474419453579127300333278, 49091717449041719016035290742176, 84772868574056134938044881265953518335, 555628412000611011592987340845035908323617024, 13889914561952086638362253697842716117160344082246744

Offset: 0

Paul D. Hanna, Nov 09 2006

nonn

			a(0) = 1 = [x^0] 1/(1-x);
a(1) = 2 = [x^1] 1/((1-x)(1-x));
a(2) = 11 = [x^2] 1/((1-x)(1-2x)(1-x));
a(3) = 184 = [x^3] 1/((1-x)(1-3x)(1-3x)(1-x));
a(4) = 10121 = [x^4] 1/((1-x)(1-4x)(1-6x)(1-4x)(1-x));
a(5) = 1911956 = [x^5] 1/((1-x)(1-5x)(1-10x)(1-10x)(1-5x)(1-x)); ...

Cf. A124834, A124835.

a[n_] := SeriesCoefficient[Product[1/(1 - Binomial[n, k]*x) , {k, 0, n}], {x, 0, n}];
Table[a[n], {n, 0, 14}] (* Jean-François Alcover, Jul 01 2017 *)

{a(n)=polcoeff(1/prod(j=0,n,1-binomial(n,j)*x +x*O(x^n)),n)}

G.f.: A(x) = [x^n] * Product_{k=0..n} 1/(1 - binomial(n,k)*x).

cp's OEIS Frontend

A124834 Triangle, read by rows, where the g.f. of column k, C_k(x), is equal to the product: C_k(x) = Product_{k=0..n} 1/(1 - binomial(n,k)*x).

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