A085881 Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by A001147(n).
1, 1, 1, 3, 6, 3, 15, 45, 45, 15, 105, 420, 630, 420, 105, 945, 4725, 9450, 9450, 4725, 945, 10395, 62370, 155925, 207900, 155925, 62370, 10395, 135135, 945945, 2837835, 4729725, 4729725, 2837835, 945945, 135135, 2027025, 16216200, 56756700, 113513400, 141891750, 113513400, 56756700, 16216200, 2027025
Offset: 0
Examples
Triangle starts:
1;
1, 1;
3, 6, 3;
15, 45, 45, 15;
105, 420, 630, 420, 105;
945, 4725, 9450, 9450, 4725, 945;
10395, 62370, 155925, 207900, 155925, 62370, 10395;
135135, 945945, 2837835, 4729725, 4729725, 2837835, 945945, 135135;
...
Links
- G. C. Greubel, Rows n = 0..100 of triangle, flattened
Programs
-
GAP
Flat(List([0..10], n-> List([0..n], k-> Binomial(n,k)*Binomial(2*n,n) *Factorial(n)/2^n ))); # G. C. Greubel, Feb 07 2020
-
Magma
[Binomial(n,k)*Binomial(2*n,n)*Factorial(n)/2^n: k in [0..n], n in [0..10]]; // G. C. Greubel, Feb 07 2020
-
Maple
T:= (n, k)-> n!/2^n*binomial(n, k)*binomial(2*n, n): seq(seq(T(n, k), k=0..n), n=0..10); # Yu-Sheng Chang, Jan 16 2020
-
Mathematica
Table[Binomial[n, k]*(2*n-1)!!, {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 07 2020 *) -
PARI
T(n,k) = binomial(n,k)*binomial(2*n,n)*n!/2^n; for(n=0,10, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Feb 07 2020
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Sage
[[binomial(n,k)*(2*n-1).multifactorial(2) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Feb 07 2020
Formula
Triangle given by [1, 2, 3, 4, 5, 6, ...] DELTA [1, 2, 3, 4, 5, 6, ...] where DELTA is Deléham's operator defined in A084938.
T(n,k) = A164961(n,k)/2^n. - Philippe Deléham, Jan 07 2012
Recurrence equation: T(n+1,k) = (2*n+1)*(T(n,k) + T(n,k-1)). - Peter Bala, Jul 15 2012
E.g.f.: 1/sqrt(1-2*x-2*x*y). - Peter Bala, Jul 15 2012
G.f.: W(0), where W(k) = 1 - (k+1)*x*(1+y)/( (k+1)*x*(1+y) - 1/W(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 03 2013