The OEIS is supported by the many generous donors to the OEIS Foundation.

 Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 59th year, we have over 358,000 sequences, and we’ve crossed 10,300 citations (which often say “discovered thanks to the OEIS”). Other ways to Give
 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A156765 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 = 1, read by rows. 3
 1, 1, 1, 1, 6, 1, 1, 42, 42, 1, 1, 360, 2520, 360, 1, 1, 3720, 223200, 223200, 3720, 1, 1, 45360, 28123200, 241056000, 28123200, 45360, 1, 1, 640080, 4839004800, 428597568000, 428597568000, 4839004800, 640080, 1, 1, 10281600, 1096841088000, 1184588375040000, 12240746542080000, 1184588375040000, 1096841088000, 10281600, 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 = 1. 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 = 1. - G. C. Greubel, Jun 19 2021 EXAMPLE Triangle begins as: 1; 1, 1; 1, 6, 1; 1, 42, 42, 1; 1, 360, 2520, 360, 1; 1, 3720, 223200, 223200, 3720, 1; 1, 45360, 28123200, 241056000, 28123200, 45360, 1; 1, 640080, 4839004800, 428597568000, 428597568000, 4839004800, 640080, 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, 1], {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, 1], {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, 1): 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, 1) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 19 2021 CROSSREFS Cf. A007318 (m=0), this sequence (m=1), A156766 (m=2), A156767 (m=3). Sequence in context: A172343 A058875 A156764 * A015117 A287020 A172375 Adjacent sequences: A156762 A156763 A156764 * A156766 A156767 A156768 KEYWORD nonn,tabl AUTHOR Roger L. Bagula, Feb 15 2009 EXTENSIONS Edited by G. C. Greubel, Jun 19 2021 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified November 29 18:13 EST 2022. Contains 358431 sequences. (Running on oeis4.)