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 A156767 Triangle T(n, k, m) = f(n, m)/(f(k, m)*f(n-k, m)), where T(0, k, m) = 1, f(n, k) = Product_{j=1..n} ( j!*((k+1)^j -1)/k ), f(n, 0) = n!, and m = 3, read by rows. 3
 1, 1, 1, 1, 10, 1, 1, 126, 126, 1, 1, 2040, 25704, 2040, 1, 1, 40920, 8347680, 8347680, 40920, 1, 1, 982800, 4021617600, 65111904000, 4021617600, 982800, 1, 1, 27523440, 2705003683200, 878482148544000, 878482148544000, 2705003683200, 27523440, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 LINKS G. C. Greubel, Rows n = 0..30 of the triangle, flattened FORMULA T(n, k, m) = f(n, m)/(f(k, m)*f(n-k, m)), where T(0, k, m) = 1, f(n, k) = Product_{j=1..n} ( j!*((k+1)^j -1)/k ), f(n, 0) = n!, and m = 3. T(n, k, m) = f(n, m)/(f(k, m)*f(n-k, m)), where T(0, k, m) = 1, f(n, k) = (-1/k)^n * BarnesG(n+2) * q-Pochhammer(k+1, k+1, n) and m = 3. - G. C. Greubel, Jun 19 2021 EXAMPLE Triangle begins as: 1; 1, 1; 1, 10, 1; 1, 126, 126, 1; 1, 2040, 25704, 2040, 1; 1, 40920, 8347680, 8347680, 40920, 1; 1, 982800, 4021617600, 65111904000, 4021617600, 982800, 1; MATHEMATICA (* First program *) b[n_, k_]:= If[k==0, n!, Product[j!*((k+1)^j -1)/k, {j, n}]]; T[n_, k_, m_]:= If[n==0, 1, b[n, m]/(b[k, m]*b[n-k, m])]; Table[T[n, k, 3], {n, 0, 12}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Jun 19 2021 *) (* Second program *) f[n_, k_]:= If[k==0, n!, (-1)^n*BarnesG[n+2] QPochhammer[k+1, k+1, n]/k^n]; T[n_, k_, m_]:= If[n==0, 1, f[n, m]/(f[k, m]*f[n-k, m])]; Table[T[n, k, 3], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Jun 19 2021 *) PROG (Magma) f:= func< n, k | n eq 0 select 1 else k eq 0 select Factorial(n) else (&*[Factorial(j)*((k+1)^j-1): j in [1..n]])/k^n >; T:= func< n, k, m | n eq 0 select 1 else f(n, m)/(f(k, m)*f(n-k, m)) >; [T(n, k, 3): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 19 2021 (Sage) from sage.combinat.q_analogues import q_pochhammer @CachedFunction def f(n, k): return factorial(n) if (k==0) else (-1/k)^n*product( factorial(j) for j in (1..n) )*q_pochhammer(n, k+1, k+1) def T(n, k, m): return 1 if (n==0) else f(n, m)/(f(k, m)*f(n-k, m)) flatten([[T(n, k, 3) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 19 2021 CROSSREFS Cf. A007318 (m=0), A156765 (m=1), A156766 (m=2), this sequence (m=3). Sequence in context: A158117 A172378 A015124 * A174921 A010180 A109013 Adjacent sequences: A156764 A156765 A156766 * A156768 A156769 A156770 KEYWORD nonn,tabl AUTHOR Roger L. Bagula, Feb 15 2009 EXTENSIONS Edited by G. C. Greubel, Jun 19 2021 STATUS approved

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Last modified February 2 03:35 EST 2023. Contains 359997 sequences. (Running on oeis4.)