A009963 Triangle of numbers n!(n-1)!...(n-k+1)!/(1!2!...k!).
1, 1, 1, 1, 2, 1, 1, 6, 6, 1, 1, 24, 72, 24, 1, 1, 120, 1440, 1440, 120, 1, 1, 720, 43200, 172800, 43200, 720, 1, 1, 5040, 1814400, 36288000, 36288000, 1814400, 5040, 1, 1, 40320, 101606400, 12192768000, 60963840000, 12192768000, 101606400, 40320, 1
Offset: 0
Examples
Rows start: 1; 1, 1; 1, 2, 1; 1, 6, 6, 1; 1, 24, 72, 24, 1; 1, 120, 1440, 1440, 120, 1; etc.
Links
- G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Crossrefs
Programs
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Magma
A009963:= func< n,k | (1/Factorial(n+1))*(&*[ Factorial(n-j+1)/Factorial(j): j in [0..k]]) >; [A009963(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jan 04 2022
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Mathematica
(* First program *) row[n_]:= Table[Product[i+j, {i,1,n-k}, {j,1,k}], {k,0,n}]; Array[row, 9, 0] // Flatten (* Jean-François Alcover, Jun 01 2019, after Peter Luschny *) (* Second program *) T[n_, k_]:= BarnesG[n+2]/(BarnesG[k+2]*BarnesG[n-k+2]); Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jan 04 2022 *) -
Sage
def A009963_row(n): return [mul(mul(i+j for j in (1..k)) for i in (1..n-k)) for k in (0..n)] for n in (0..7): A009963_row(n) # Peter Luschny, Nov 26 2012
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Sage
def triangle_to_n_rows(n): #changing n will give you the triangle to row n. N=[[1]+n*[0]] for i in [1..n]: N.append([]) for j in [0..n]: if i>=j: N[i].append(factorial(i-j)*binomial(i-1,j-1)*N[i-1][j-1]+factorial(j)*binomial(i-1,j)*N[i-1][j]) else: N[i].append(0) return [[N[i][j] for j in [0..i]] for i in [0..n]] # Tom Edgar, Feb 13 2014
Formula
T(n,k) = T(n-1,k-1)*A008279(n,n-k) = A000178(n)/(A000178(k)*A000178(n-k)) i.e., a "supercombination" of "superfactorials". - Henry Bottomley, May 22 2002
Equals ConvOffsStoT transform of the factorials starting (1, 2, 6, 24, ...); e.g., ConvOffs transform of (1, 2, 6, 24) = (1, 24, 72, 24, 1). Note that A090441 = ConvOffsStoT transform of the factorials, A000142. - Gary W. Adamson, Apr 21 2008
Asymptotic: T(n,k) ~ exp((3/2)*k^2 - zeta'(-1) + 3/4 - (3/2)*n*k)*(1+n)^((1/2)*n^2 + n + 5/12)*(1+k)^(-(1/2)*k^2 - k - 5/12)*(1 + n - k)^(-(1/2)*n^2 + n*k - (1/2)*k^2 - n + k - 5/12)/(sqrt(2*Pi). - Peter Luschny, Nov 26 2012
T(n,k) = (n-k)!*C(n-1,k-1)*T(n-1,k-1) + k!*C(n-1,k)*T(n-1,k) where C(i,j) is given by A007318. - Tom Edgar, Feb 13 2014
T(n,k) = Product_{i=1..k} (n+1-i)!/i!. - Alois P. Heinz, Jun 07 2017
T(n,k) = BarnesG(n+2)/(BarnesG(k+2)*BarnesG(n-k+2)). - G. C. Greubel, Jan 04 2022
Comments