login
A171814
Triangle T : T(n,k)= A007318(n,k)*A001700(n-k).
0
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
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 *)
KEYWORD
nonn,tabl
AUTHOR
Philippe Deléham, Dec 19 2009
STATUS
approved