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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 (list; table; graph; refs; listen; history; text; internal format)
 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: A317548 A320327 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

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Last modified September 22 11:05 EDT 2023. Contains 365520 sequences. (Running on oeis4.)