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A085881 Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by A001147(n).

%I A085881 #31 Sep 08 2022 08:45:11
%S A085881 1,1,1,3,6,3,15,45,45,15,105,420,630,420,105,945,4725,9450,9450,4725,
%T A085881 945,10395,62370,155925,207900,155925,62370,10395,135135,945945,
%U A085881 2837835,4729725,4729725,2837835,945945,135135,2027025,16216200,56756700,113513400,141891750,113513400,56756700,16216200,2027025
%N A085881 Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by A001147(n).
%H A085881 G. C. Greubel, <a href="/A085881/b085881.txt">Rows n = 0..100 of triangle, flattened</a>
%F A085881 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.
%F A085881 T(n,k) = A164961(n,k)/2^n. - _Philippe Deléham_, Jan 07 2012
%F A085881 Recurrence equation: T(n+1,k) = (2*n+1)*(T(n,k) + T(n,k-1)). - _Peter Bala_, Jul 15 2012
%F A085881 E.g.f.: 1/sqrt(1-2*x-2*x*y). - _Peter Bala_, Jul 15 2012
%F A085881 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
%e A085881 Triangle starts:
%e A085881        1;
%e A085881        1,      1;
%e A085881        3,      6,       3;
%e A085881       15,     45,      45,      15;
%e A085881      105,    420,     630,     420,     105;
%e A085881      945,   4725,    9450,    9450,    4725,     945;
%e A085881    10395,  62370,  155925,  207900,  155925,   62370,  10395;
%e A085881   135135, 945945, 2837835, 4729725, 4729725, 2837835, 945945, 135135;
%e A085881   ...
%p A085881 T:= (n, k)-> n!/2^n*binomial(n, k)*binomial(2*n, n):
%p A085881 seq(seq(T(n, k), k=0..n), n=0..10); # _Yu-Sheng Chang_, Jan 16 2020
%t A085881 Table[Binomial[n, k]*(2*n-1)!!, {n,0,10}, {k,0,n}]//Flatten (* _G. C. Greubel_, Feb 07 2020 *)
%o A085881 (PARI) T(n,k) = binomial(n,k)*binomial(2*n,n)*n!/2^n;
%o A085881 for(n=0,10, for(k=0,n, print1(T(n,k), ", "))) \\ _G. C. Greubel_, Feb 07 2020
%o A085881 (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
%o A085881 (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
%o A085881 (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
%Y A085881 Cf. A001147, A007318, A084938, A164961.
%K A085881 nonn,tabl
%O A085881 0,4
%A A085881 _N. J. A. Sloane_, Aug 17 2003