A171814 Triangle T : T(n,k)= A007318(n,k)*A001700(n-k).

1, 3, 1, 10, 6, 1, 35, 30, 9, 1, 126, 140, 60, 12, 1, 462, 630, 350, 100, 15, 1, 1716, 2772, 1890, 700, 150, 18, 1, 6435, 12012, 9702, 4410, 1225, 210, 21, 1, 24310, 51480, 48048, 25872, 8820, 1960, 280, 24, 1

Offset

0,2

Links

Table of n, a(n) for n=0..44.

Formula

Sum_{k, 0<=k<=n} T(n,k)*x^k = A168491(n), A099323(n+1), A001405(n), A005773(n+1), A001700(n), A026378(n+1), A005573(n), A122898(n) for x = -4, -3, -2, -1, 0, 1, 2, 3 respectively.

Conjectural g.f.: 1/(2*t)*( sqrt( (1 - x*t)/(1 - (4 + x)*t) ) - 1 ) = 1 + (3 + x)*t + (10 + 6*x + x^2)*t^2 + .... - Peter Bala, Nov 10 2013

E.g.f. of column k: exp(2*x)*(BesselI(0,2*x)+BesselI(1,2*x))*x^k / k!. - Mélika Tebni, Dec 23 2023

Example

Triangle begins:
     1;
     3,    1;
    10,    6,    1;
    35,   30,    9,   1;
   126,  140,   60,  12,   1;
   462,  630,  350, 100,  15,  1;
  1716, 2772, 1890, 700, 150, 18, 1;
  ...

Mathematica

T[n_, k_]:=n!SeriesCoefficient[Exp[2*x]*(BesselI[0, 2*x]+BesselI[1, 2*x])*x^k / k!, {x, 0, n}]; Table[T[n, k], {n, 0, 8}, {k, 0, n}]//Flatten (* Stefano Spezia, Dec 23 2023 *)

Crossrefs

Cf. A107230, A171651

Cf. A001405, A001700, A005573, A005773, A026378, A099323, A122898, A168491.

Sequence in context: A171505 A134283 A035324 * A091965 A171568 A107056

Adjacent sequences: A171811 A171812 A171813 * A171815 A171816 A171817

Keyword

nonn,tabl

Author

Philippe Deléham, Dec 19 2009

Status

approved