A153274 - OEIS

%I #7 Sep 08 2022 08:45:39

%S 2,6,15,24,105,280,120,945,3640,9945,720,10395,58240,208845,576576,

%T 5040,135135,1106560,5221125,17873856,49579075,40320,2027025,24344320,

%U 151412625,643458816,2131900225,5925744000,362880,34459425,608608000,4996616625,26381811456,104463111025,337767408000,939536222625

%N Triangle, read by rows, T(n,k) = k^(n+1) * Pochhammer(1/k, n+1).

%C A Pochhammer function-based triangular sequence.

%C Row sums are: {2, 21, 409, 14650, 854776, 73920791, 8878927331, 1413788600036, 288152651134776, 73152069870215127, ...}.

%H G. C. Greubel, <a href="/A153274/b153274.txt">Rows n = 1..100 of triangle, flattened</a>

%F T(n, k) = k^(n+1) * Pochmammer(1/k, n+1).

%F T(n, k) = Product_{j=0..n} (j*k + 1). - _G. C. Greubel_, Mar 05 2020

%e Triangle begins as:

%e       2;

%e       6,      15;

%e      24,     105,      280;

%e     120,     945,     3640,      9945;

%e     720,   10395,    58240,    208845,    576576;

%e    5040,  135135,  1106560,   5221125,  17873856,   49579075;

%e   40320, 2027025, 24344320, 151412625, 643458816, 2131900225, 5925744000;

%p seq(seq( k^(n+1)*pochhammer(1/k, n+1), k=1..n), n=1..12); # _G. C. Greubel_, Mar 05 2020

%t Table[Apply[Plus, CoefficientList[j*k^n*Pochhammer[(j+k)/k, n], j]], {n, 12}, {k,n}]//Flatten (* modified by _G. C. Greubel_, Mar 05 2020 *)

%t Table[k^(n+1)*Pochhammer[1/k, n+1], {n,12}, {k,n}]//Flatten (* _G. C. Greubel_, Mar 05 2020 *)

%o (PARI) T(n, k) = prod(j=0, n, j*k+1);

%o for(n=1, 12, for(k=1, n, print1(T(n, k), ", "))) \\ _G. C. Greubel_, Mar 05 2020

%o (Magma) [(&*[j*k+1: j in [0..n]]): k in [1..n], n in [1..12]]; // _G. C. Greubel_, Mar 05 2020

%o (Sage) [[k^(n+1)*rising_factorial(1/k,n+1) for k in (1..n)] for n in (1..12)] # _G. C. Greubel_, Mar 05 2020

%o (GAP) Flat(List([1..12], n-> List([1..n], k-> Product([0..n], j-> j*k+1 )))); # _G. C. Greubel_, Mar 05 2020

%Y Cf. A000142, A001147, A007559, A007696, A008548, A008542, A045754, A045755, A045756.

%Y Cf. A142589, A256268.

%K nonn,tabl

%O 1,1

%A _Roger L. Bagula_, Dec 22 2008

%E Edited by _G. C. Greubel_, Mar 05 2020