A153189 Triangle T(n,k) = Product_{j=0..k} n*j+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

Links

G. C. Greubel, Rows n = 0..100 of triangle, flattened

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. A000165, A006882, A007661, A007662, A008544.

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

Adjacent sequences: A153186 A153187 A153188 * A153190 A153191 A153192

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