A183875 Triangle T(n,k) for A(x)^k=sum(n>=k T(n,k)*x^n), where o.g.f. A(x) satisfies A(x)=(a+b*x*A(x))/(c-d*x*A(x)), a=1,b=2,c=1,d=2.

1, 4, 1, 24, 8, 1, 176, 64, 12, 1, 1440, 544, 120, 16, 1, 12608, 4864, 1168, 192, 20, 1, 115584, 45184, 11424, 2112, 280, 24, 1, 1095424, 432128, 113088, 22528, 3440, 384, 28, 1, 10646016, 4227584, 1133952, 237824, 39840, 5216, 504, 32, 1, 105522176, 42115072, 11506944, 2505728, 448064, 65280, 7504, 640, 36, 1

Offset

1,2

Comments

For o.g.f G(x), G(A(x,a,b,c,d))=g(0)+sum(n>0, sum(k=1..n, T(n,k,a,b,c,d)*g(k))x^n).

T(n,k,1,1,1,1)=A080247(n,k),

T(n,k,2,-1,1,1)=A108891(n,k),

T(n,k,1,-2,1,1)=A125692(n,k),

T(n,k,1,-3,1,1)=A125694(n,k),

T(n,k,-2,1,1,1)=A085403(n,k).

Links

Table of n, a(n) for n=1..55.

Vladimir Kruchinin, D. V. Kruchinin, Composita and their properties, arXiv:1103.2582 [math.CO], 2011-2013.

Formula

T(n,k,a,b,c,d):=k/n*sum(i=0..n-k, binomial(n,n-k-i)*a^(k+i)*b^(n-k-i)*binomial(i+n-1,n-1)*c^(-i-n)*d^i), a,b,c,d !=0, n>0.

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

Conjecture: T(n,1) = A156017(n-1). - R. J. Mathar, Nov 14 2011

Example

1,
4,1,
24,8,1,
176,64,12,1,
1440,544,120,16,1,
12608,4864,1168,192,20,1,
115584,45184,11424,2112,280,24,1,
1095424,432128,113088,22528,3440,384,28,1,
10646016,4227584,1133952,237824,39840,5216,504,32,1,
105522176,42115072,11506944,2505728,448064,65280,7504,640,36,1

Mathematica

T[n_, k_, a_, b_, c_, d_] := k/n Sum[Binomial[n, n - k - i] a^(k + i) b^(n - k - i) Binomial[i + n - 1, n - 1] c^(-i - n) d^i, {i, 0, n - k}];
T[n_, k_] := T[n, k, 1, 2, 1, 2];
Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Aug 08 2018, from formula *)

Crossrefs

Sequence in context: A158978 A128417 A257532 * A136232 A079621 A285061

Adjacent sequences: A183872 A183873 A183874 * A183876 A183877 A183878

Keyword

nonn,tabl

Author

Vladimir Kruchinin, Feb 12 2011

Status

approved