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A153189
Triangle T(n,k) = Product_{j=0..k} n*j+1.
1
1, 1, 2, 1, 3, 15, 1, 4, 28, 280, 1, 5, 45, 585, 9945, 1, 6, 66, 1056, 22176, 576576, 1, 7, 91, 1729, 43225, 1339975, 49579075, 1, 8, 120, 2640, 76560, 2756160, 118514880, 5925744000, 1, 9, 153, 3825, 126225, 5175225, 253586025, 14454403425, 939536222625
OFFSET
0,3
COMMENTS
Row sums are: {1, 3, 19, 313, 10581, 599881, 50964103, 6047094369, 954249517513, 193146844030201, 48762935887310811,...}. [Corrected by M. F. Hasler, Oct 28 2014]
This is the lower left triangle of the array A142589. - M. F. Hasler, Oct 28 2014
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
FORMULA
T(n, k) = n^(k+1)*Pochhammer(1/n, k+1).
From Vaclav Kotesovec, Oct 10 2016: (Start)
For fixed n > 0:
T(n, k) ~ sqrt(2*Pi) * n^k * k^(k + 1/2 + 1/n) / (Gamma(1 + 1/n) * exp(k)).
T(n, k) ~ k! * n^k * k^(1/n) / Gamma(1 + 1/n).
(End)
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
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
EXAMPLE
Triangle begins as:
1;
1, 2;
1, 3, 15;
1, 4, 28, 280;
1, 5, 45, 585, 9945;
1, 6, 66, 1056, 22176, 576576;
1, 7, 91, 1729, 43225, 1339975, 49579075;
1, 8, 120, 2640, 76560, 2756160, 118514880, 5925744000;
1, 9, 153, 3825, 126225, 5175225, 253586025, 14454403425, 939536222625;
MAPLE
seq(seq(mul(n*j+1, j=0..k), k=0..n), n=0..10); # G. C. Greubel, Feb 15 2020
MATHEMATICA
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[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 *)
PROG
(PARI) T(n, k)=prod(j=0, k, n*j+1) \\ M. F. Hasler, Oct 28 2014
(Magma) [(&*[n*j+1: j in [0..k]]): k in [0..n], n in [0..10]]; // G. C. Greubel, Feb 15 2020
(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
CROSSREFS
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).
Sequence in context: A320327 A366591 A007447 * A362272 A095852 A283748
KEYWORD
nonn,tabl
AUTHOR
Roger L. Bagula, Dec 20 2008
EXTENSIONS
Edited and row 0 added by M. F. Hasler, Oct 28 2014
STATUS
approved