A027307 -id:A027307 - cp's OEIS frontend

A257532 Triangle, read by rows, T(n,k)=k/nSum_{i=0..n-k} C(2n,n-k-i)C(2n+i-1,i).

1, 4, 1, 24, 8, 1, 172, 64, 12, 1, 1360, 536, 120, 16, 1, 11444, 4672, 1156, 192, 20, 1, 100520, 42024, 11088, 2096, 280, 24, 1, 911068, 387456, 106908, 22016, 3420, 384, 28, 1, 8457504, 3643448, 1038984, 227408, 39120, 5192, 504, 32, 1

Offset: 1

Vladimir Kruchinin, Apr 28 2015

			1;
4, 1;
24, 8, 1;
172, 64, 12, 1;
1360, 536, 120, 16, 1;

Cf. A027307. First column = A032349.

T(n,k):=(k*sum(binomial(2*n,n-k-i)*binomial(2*n+i-1,i),i,0,n-k))/n;

G.f.: 1/(1-x*B(x)^2*y)-1, where B(x) is g.f. of A027307.

G.f. satisfies A(x)=x*[(1+A(x))/(1-A(x))]^2.

A305537 G.f. A(x) satisfies: A(x) = 1/(1 - xA(x) - xA(x)/(1 - xA(x) - 2xA(x)/(1 - xA(x) - 3xA(x)/(1 - xA(x) - 4x*A(x)/(1 - ...))))), a continued fraction.

Original entry on oeis.org

1, 2, 11, 87, 844, 9438, 118217, 1636078, 24869591, 414422424, 7568815758, 151468591827, 3317061005044, 79265498450882, 2058189152006115, 57777549430984983, 1744191365957251044, 56332730020388347302, 1937412176139535240463, 70659708678402399722656

Offset: 0

Ilya Gutkovskiy, Jun 04 2018

nonn

			G.f. A(x) = 1 + 2*x + 11*x^2 + 87*x^3 + 844*x^4 + 9438*x^5 + 118217*x^6 + 1636078*x^7 + 24869591*x^8 + ...

Index entries for sequences related to Bessel functions or polynomials

Cf. A001515, A027307, A301363.

a(n) = [x^n] (Sum_{k>=0} A001515(k)*x^k)^(n+1)/(n + 1).

cp's OEIS Frontend

A257532 Triangle, read by rows, T(n,k)=k/nSum_{i=0..n-k} C(2n,n-k-i)C(2n+i-1,i).

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A305537 G.f. A(x) satisfies: A(x) = 1/(1 - xA(x) - xA(x)/(1 - xA(x) - 2xA(x)/(1 - xA(x) - 3xA(x)/(1 - xA(x) - 4x*A(x)/(1 - ...))))), a continued fraction.

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cp's OEIS Frontend

A257532 Triangle, read by rows, T(n,k)=k/n*Sum_{i=0..n-k} C(2*n,n-k-i)*C(2*n+i-1,i).

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Maxima

Formula

A305537 G.f. A(x) satisfies: A(x) = 1/(1 - x*A(x) - x*A(x)/(1 - x*A(x) - 2*x*A(x)/(1 - x*A(x) - 3*x*A(x)/(1 - x*A(x) - 4*x*A(x)/(1 - ...))))), a continued fraction.

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Formula

A257532 Triangle, read by rows, T(n,k)=k/nSum_{i=0..n-k} C(2n,n-k-i)C(2n+i-1,i).

A305537 G.f. A(x) satisfies: A(x) = 1/(1 - xA(x) - xA(x)/(1 - xA(x) - 2xA(x)/(1 - xA(x) - 3xA(x)/(1 - xA(x) - 4x*A(x)/(1 - ...))))), a continued fraction.