A153189 - OEIS

%I #34 Sep 08 2022 08:45:39

%S 1,1,2,1,3,15,1,4,28,280,1,5,45,585,9945,1,6,66,1056,22176,576576,1,7,

%T 91,1729,43225,1339975,49579075,1,8,120,2640,76560,2756160,118514880,

%U 5925744000,1,9,153,3825,126225,5175225,253586025,14454403425,939536222625

%N Triangle T(n,k) = Product_{j=0..k} n*j+1.

%C Row sums are: {1, 3, 19, 313, 10581, 599881, 50964103, 6047094369, 954249517513, 193146844030201, 48762935887310811,...}. [Corrected by _M. F. Hasler_, Oct 28 2014]

%C This is the lower left triangle of the array A142589. - _M. F. Hasler_, Oct 28 2014

%C Row n is a subset of the n-fold factorial sequence for k=0..n. For example, T(8,0..8) is A045755(1..9).  These sequences are listed for n=0..10 in A256268. - _Georg Fischer_, Feb 15 2020

%H G. C. Greubel, <a href="/A153189/b153189.txt">Rows n = 0..100 of triangle, flattened</a>

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

%F From _Vaclav Kotesovec_, Oct 10 2016: (Start)

%F For fixed n > 0:

%F T(n, k) ~ sqrt(2*Pi) * n^k * k^(k + 1/2 + 1/n) / (Gamma(1 + 1/n) * exp(k)).

%F T(n, k) ~ k! * n^k * k^(1/n) / Gamma(1 + 1/n).

%F (End)

%F T(n, k) = Sum_{j=0..k+1} (-1)^(k-j+1)*Stirling1(k+1,j)*n^(k-j+1). - _G. C. Greubel_, Feb 17 2020

%F T(n, k) = ((1+n*k)*T(n, k-1) + (1+n*k)*(1+n*(k-1))*T(n, k-2))/2. - _Georg Fischer_, Feb 17 2020

%e Triangle begins as:

%e   1;

%e   1, 2;

%e   1, 3,  15;

%e   1, 4,  28,  280;

%e   1, 5,  45,  585,   9945;

%e   1, 6,  66, 1056,  22176,  576576;

%e   1, 7,  91, 1729,  43225, 1339975,  49579075;

%e   1, 8, 120, 2640,  76560, 2756160, 118514880,  5925744000;

%e   1, 9, 153, 3825, 126225, 5175225, 253586025, 14454403425, 939536222625;

%p seq(seq(mul(n*j+1, j=0..k), k=0..n), n=0..10); # _G. C. Greubel_, Feb 15 2020

%t T[n_, k_]= If[n==0 && k==0, 1, Product[n*j+1, {j,0,k}]]; Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten (* _G. C. Greubel_, Feb 15 2020 *)

%t T[n_, k_]:= T[n, k]= If[k<2, 1+k*n, ((1+n*k)*T[n, k-1] + (1+n*k)*(1+n*(k-1))* T[n, k-2])/2]; Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten (* _Georg Fischer_, Feb 17 2020 *)

%o (PARI) T(n,k)=prod(j=0,k,n*j+1) \\ _M. F. Hasler_, Oct 28 2014

%o (Magma) [(&*[n*j+1: j in [0..k]]): k in [0..n], n in [0..10]]; // _G. C. Greubel_, Feb 15 2020

%o (Sage) [[ product(n*j+1 for j in (0..k)) for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, Feb 15 2020

%Y Cf. A000165, A006882, A007661, A007662, A008544.

%Y Cf. A000142 (row 2), A001147 (3), A007559 (4), A007696 (5), A008548 (6), A008542 (7), A045754 (8), A045755 (9), A045756 (10), A144773 (11), A256268 (combined table).

%K nonn,tabl

%O 0,3

%A _Roger L. Bagula_, Dec 20 2008

%E Edited and row 0 added by _M. F. Hasler_, Oct 28 2014