A154914 - OEIS

%I #13 Mar 02 2021 02:07:06

%S 4,5,5,13,-24,13,35,-30,-30,35,97,-936,1584,-936,97,275,2940,-2700,

%T -2700,2940,275,793,-78570,168012,-194400,168012,-78570,793,2315,

%U 1153350,-2002140,960120,960120,-2002140,1153350,2315,6817,-24113544,46757880,-42378336,35090496,-42378336,46757880,-24113544,6817

%N Triangle T(n,m,p,q) = (p^(n-k)*q^k + p^k*q^(n-k))*(StirlingS1(n, k) + StirlingS1(n, n-k)) with p=2 and q=3, read by rows.

%H G. C. Greubel, <a href="/A154914/b154914.txt">Rows n = 0..50 of the triangle, flattened</a>

%F T(n,m,p,q) = (p^(n-k)*q^k + p^k*q^(n-k))*(StirlingS1(n, k) + StirlingS1(n, n-k)) with p=2 and q=3.

%F Sum_{k=0..n} T(n,k,p,q) = 2*(-p)^n*Pochhammer(-q/p, n) + p^(n+1)*[n < 2], where p=2 and q=3. - _G. C. Greubel_, Mar 02 2021

%e Triangle begins as:

%e      4;

%e      5,       5;

%e     13,     -24,       13;

%e     35,     -30,      -30,      35;

%e     97,    -936,     1584,    -936,     97;

%e    275,    2940,    -2700,   -2700,   2940,      275;

%e    793,  -78570,   168012, -194400, 168012,   -78570,     793;

%e   2315, 1153350, -2002140,  960120, 960120, -2002140, 1153350, 2315;

%p A154914:= (n,k,p,q) -> (p^(n-k)*q^k + p^k*q^(n-k))*(combinat[stirling1](n, k) + combinat[stirling1](n, n-k));

%p seq(seq(A154914(n,k,2,3), k=0..n), n=0..12); # _G. C. Greubel_, Mar 02 2021

%t T[n_, k_, p_, q_]:= (p^(n-k)*q^k + p^k*q^(n-k))*(StirlingS1[n, k] + StirlingS1[n, n-k]);

%t Table[T[n, k, 2, 3], {n,0,12}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Mar 02 2021 *)

%o (Sage)

%o def A154914(n,k,p,q): return (p^(n-k)*q^k + p^k*q^(n-k))*(stirling_number1(n, k) + stirling_number1(n, n-k))

%o flatten([[A154914(n,k,2,3) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Mar 02 2021

%o (Magma)

%o A154914:= func< n,k,p,q | (p^(n-k)*q^k + p^k*q^(n-k))*(StirlingFirst(n, k) + StirlingFirst(n, n-k)) >;

%o [A154914(n,k,2,3): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Mar 02 2021

%Y Cf. A154913 (q=1), this sequence (q=3).

%Y Cf. A048994, A154915, A154916, A154922.

%K tabl,sign,easy,less

%O 0,1

%A _Roger L. Bagula_, Jan 17 2009

%E Edited by _G. C. Greubel_, Mar 02 2021