A009963 -id:A009963 - cp's OEIS frontend

A156584 Triangle T(n,k) = SF(n+1)/(SF(n-k+1)*SF(k+1)) where SF(n) is the superfactorial A000178(n), read by rows.

1, 1, 1, 1, 3, 1, 1, 12, 12, 1, 1, 60, 240, 60, 1, 1, 360, 7200, 7200, 360, 1, 1, 2520, 302400, 1512000, 302400, 2520, 1, 1, 20160, 16934400, 508032000, 508032000, 16934400, 20160, 1, 1, 181440, 1219276800, 256048128000, 1536288768000, 256048128000, 1219276800, 181440, 1

Offset: 0

Roger L. Bagula, Feb 10 2009

			Triangle begins as:
  1;
  1,     1;
  1,     3,        1;
  1,    12,       12,         1;
  1,    60,      240,        60,         1;
  1,   360,     7200,      7200,       360,        1;
  1,  2520,   302400,   1512000,    302400,     2520,     1;
  1, 20160, 16934400, 508032000, 508032000, 16934400, 20160, 1;

G. C. Greubel, Rows n = 0..30 of the triangle, flattened

Cf. A007318 (m=0), this sequence (m=1), A156764 (m=3).

Cf. A009963.

SF := n -> mul(j!, j=0..n): T := (n,k) -> SF(n-1)/(SF(n-k)*SF(k)):
seq(print(seq(T(n,k),k=1..n-1)),n=0..9); # Peter Luschny, Jan 24 2015

(* First program *)
b[n_, k_]:= If[k==0, n!, Product[Sum[(-1)^(i+j)*(j+1)*StirlingS1[j-1, i]*(k+1)^i, {i, 0, j-1}], {j, 1, 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 20 2021 *)
(* Second program *)
f[n_, k_]:= If[k==0, n!, (-1)^n*(n+1)!*BarnesG[n+k+1]/(Gamma[k+1]^n*BarnesG[k+1])];
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 20 2021 *)

def f(n,k): return factorial(n) if (k==0) else (-1)^n*factorial(n+1)*product( rising_factorial(k+1, j) for j in (0..n-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 21 2021

From G. C. Greubel, Jun 21 2021: (Start)
T(n, k) = BarnesG(n+3)/(BarnesG(k+3)*BarnesG(n-k+3)).
T(n, k, m) = f(n, m)/(f(k, m)*f(n-k, m)), with T(0, k, m) = 1, f(n, k) = (-1)^n*(n + 1)!*BarnesG(n+k+1)/(Gamma(k+1)^n*BarnesG(k+1)), f(n, 0) = n!, and m = 1. (End)

New name and editing, Peter Luschny, Jan 24 2015

A193521 G.f.: A(x) = ( Sum_{n>=0} x^n/sf(n) )^3 where A(x) = Sum_{n>=0} a(n)*x^n/sf(n), and sf(n) = Product_{k=0..n} k! is the superfactorial of n (A000178).

Original entry on oeis.org

1, 3, 9, 51, 795, 43923, 10372323, 11996843043, 75315947454723, 2788806652875290883, 654625444656522114316803, 1045012738906587147509753740803, 12046169853230117709495421609499289603, 1053916215003128938522329980606467994425804803

Offset: 0

Paul D. Hanna, Jul 29 2011

nonn

			Let F(x) = 1 + x + x^2/(1!*2!) + x^3/(1!*2!*3!) + x^4/(1!*2!*3!*4!) + ... + x^n/sf(n) + ...
then F(x)^3 = 1 + 3*x + 9*x^2/(1!*2!) + 51*x^3/(1!*2!*3!) + 795*x^4/(1!*2!*3!*4!) + 43923*x^5/(1!*2!*3!*4!*5!) + ... + a(n)*x^n/sf(n) + ...

G. C. Greubel, Table of n, a(n) for n = 0..48

Cf. A000178, A009963, A193520.

A193521:= func< n | (&+[ A009963(n,k)*A193520(k): k in [0..n]]) >;
[A193521(n): n in [0..20]]; // G. C. Greubel, Jan 05 2022

a[n_]:= a[n]= Sum[BarnesG[n+2]/(BarnesG[j+2]*BarnesG[k-j+2]*BarnesG[n-k+2]), {k,0,n}, {j,0,k}];
Table[a[n], {n, 0, 20}] (* G. C. Greubel, Jan 05 2022 *)

{a(n) = prod(k=1,n,k!)*polcoeff((sum(m=0, n+1, x^m/prod(k=0, m, k!) + x*O(x^n))^3), n)}

@CachedFunction
def A009963(n,k): return product(factorial(n-j+1)/factorial(j) for j in (1..k))
def A193521(n): return sum(sum(A009963(n,k)*A009963(k,j) for j in (0..k)) for k in (0..n))
[A193521(n) for n in (0..20)] # G. C. Greubel, Jan 05 2022

From G. C. Greubel, Jan 05 2022: (Start)
a(n) = Sum_{k=0..n} Sum_{j=0..k} BarnesG(n+2)/(BarnesG(j+2)*BarnesG(k-j+2 )*BarnesG(n-k+2)).
a(n) = Sum_{k=0..n} A009963(n, k) * Sum_{j=0..k} A009963(k, j).
a(n) = Sum_{j=0..n} A009963(n, j)*A193520(j). (End)
a(n) ~ c(n) * A^2 * 3^(5/4 + n + n^2/6) * n^(-5/6 + n^2/3) / (2*Pi * exp(1/6 + n^2/2)), where c(n) = 1 if mod(n,3) = 0 and c(n) = 3^(4/3) / n^(1/3) if mod(n,3) = 1 or if mod(n,3) = 2, A = A074962 is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Aug 29 2023

A156586 A new q-combination type general triangle sequence based on Stirling first polynomials: here q=4: m=3: t(n,k)=If[m == 0, n!, Product[Sum[(-1)^(i + k)StirlingS1[k - 1, i](m + 1)^i, {i, 0, k - 1}], {k, 1, n}]]; b(n,k,m)=If[n == 0, 1, t[n, m]/(t[k, m]*t[n - k, m])].

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 20, 20, 1, 1, 120, 600, 120, 1, 1, 840, 25200, 25200, 840, 1, 1, 6720, 1411200, 8467200, 1411200, 6720, 1, 1, 60480, 101606400, 4267468800, 4267468800, 101606400, 60480, 1, 1, 604800, 9144576000, 3072577536000, 21508042752000

Offset: 0

Roger L. Bagula, Feb 10 2009

Row sums are:
{1, 2, 6, 42, 842, 52082, 11303042, 8738271362, 27671488185602,
346773112532985602, 20244862147392528307202,...}.
The q=2 sequence is A009963.

			{1},
{1, 1},
{1, 4, 1},
{1, 20, 20, 1},
{1, 120, 600, 120, 1},
{1, 840, 25200, 25200, 840, 1},
{1, 6720, 1411200, 8467200, 1411200, 6720, 1},
{1, 60480, 101606400, 4267468800, 4267468800, 101606400, 60480, 1},
{1, 604800, 9144576000, 3072577536000, 21508042752000, 3072577536000, 9144576000, 604800, 1},
{1, 6652800, 1005903360000, 3041851760640000, 170343698595840000, 170343698595840000, 3041851760640000, 1005903360000, 6652800, 1},
{1, 79833600, 132779243520000, 4015244324044800000, 2023683139318579200000, 16189465114548633600000, 2023683139318579200000, 4015244324044800000, 132779243520000, 79833600, 1}

A009963

Clear[t, n, m, i, k, a, b];
t[n_, m_] = If[m == 0, n!, Product[Sum[(-1)^(i + k)*StirlingS1[k - 1, i]*(m + 1)^i, {i, 0, k - 1}], {k, 1, n}]];
b[n_, k_, m_] = If[n == 0, 1, t[n, m]/(t[k, m]*t[n - k, m])];
Table[Flatten[Table[Table[b[n, k, m], {k, 0, n}], {n, 0, 10}]], {m, 0, 15}]

q=4: m=3:
t(n,k)=If[m == 0, n!, Product[Sum[(-1)^(i + k)*StirlingS1[k - 1, i]*(m + 1)^i, {i, 0, k - 1}], {k, 1, n}]];
b(n,k,m)=If[n == 0, 1, t[n, m]/(t[k, m]*t[n - k, m])].

A156587 A new q-combination type general triangle sequence based on Stirling first polynomials: here q=5: m=4: t(n,k)=If[m == 0, n!, Product[Sum[(-1)^(i + k)StirlingS1[k - 1, i](m + 1)^i, {i, 0, k - 1}], {k, 1, n}]]; b(n,k,m)=If[n == 0, 1, t[n, m]/(t[k, m]*t[n - k, m])].

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 1, 30, 30, 1, 1, 210, 1260, 210, 1, 1, 1680, 70560, 70560, 1680, 1, 1, 15120, 5080320, 35562240, 5080320, 15120, 1, 1, 151200, 457228800, 25604812800, 25604812800, 457228800, 151200, 1, 1, 1663200, 50295168000, 25348764672000

Offset: 0

Roger L. Bagula, Feb 10 2009

Row sums are:
{1, 2, 7, 62, 1682, 144482, 45753122, 52124385602, 253588240382402,
4885227205552108802, 454865349223042267910402,...}.
The q=2 sequence is A009963.

			{1},
{1, 1},
{1, 5, 1},
{1, 30, 30, 1},
{1, 210, 1260, 210, 1},
{1, 1680, 70560, 70560, 1680, 1},
{1, 15120, 5080320, 35562240, 5080320, 15120, 1},
{1, 151200, 457228800, 25604812800, 25604812800, 457228800, 151200, 1},
{1, 1663200, 50295168000, 25348764672000, 202790117376000, 25348764672000, 50295168000, 1663200, 1},
{1, 19958400, 6638962176000, 33460369367040000, 2409146594426880000, 2409146594426880000, 33460369367040000, 6638962176000, 19958400, 1},
{1, 259459200, 1035678099456000, 57417993833840640000, 41340955560365260800000, 372068600043287347200000, 41340955560365260800000, 57417993833840640000, 1035678099456000, 259459200, 1}

A009963

Clear[t, n, m, i, k, a, b];
t[n_, m_] = If[m == 0, n!, Product[Sum[(-1)^(i + k)*StirlingS1[k - 1, i]*(m + 1)^i, {i, 0, k - 1}], {k, 1, n}]];
b[n_, k_, m_] = If[n == 0, 1, t[n, m]/(t[k, m]*t[n - k, m])];
Table[Flatten[Table[Table[b[n, k, m], {k, 0, n}], {n, 0, 10}]], {m, 0, 15}]

q=5: m=4:
t(n,k)=If[m == 0, n!, Product[Sum[(-1)^(i + k)*StirlingS1[k - 1, i]*(m + 1)^i, {i, 0, k - 1}], {k, 1, n}]];
b(n,k,m)=If[n == 0, 1, t[n, m]/(t[k, m]*t[n - k, m])].

A156588 A triangle of q factorial type based on Stirling first polynomials: t(n,k)=If[m == 0, n!, Product[Sum[(-1)^(i + k)StirlingS1[k - 1, i](m + 1)^i, {i, 0, k - 1}], {k, 1, n}]].

Original entry on oeis.org

1, 1, 1, 1, -1, 2, 1, -1, 2, 6, 1, -1, 3, -12, 24, 1, -1, 4, -36, 288, 120, 1, -1, 5, -80, 2160, -34560, 720, 1, -1, 6, -150, 9600, -777600, 24883200, 5040, 1, -1, 7, -252, 31500, -8064000, 1959552000, -125411328000, 40320, 1, -1, 8, -392, 84672, -52920000

Offset: 0

Roger L. Bagula, Feb 10 2009

Row sums are:
{1, 2, 2, 8, 15, 376, -31755, 24120096, -123459768425, 5017134314247168,
-1827769039991244222327,...}.

			{1},
{1, 1},
{1, -1, 2},
{1, -1, 2, 6},
{1, -1, 3, -12, 24},
{1, -1, 4, -36, 288, 120},
{1, -1, 5, -80, 2160, -34560, 720},
{1, -1, 6, -150, 9600, -777600, 24883200, 5040},
{1, -1, 7, -252, 31500, -8064000, 1959552000, -125411328000, 40320},
{1, -1, 8, -392, 84672, -52920000, 54190080000, -39504568320000, 5056584744960000, 362880},
{1, -1, 9, -576, 197568, -256048128, 800150400000, -3277416038400000, 7167708875980800000, -1834933472251084800000, 3628800}

A009963

Clear[t, n, m, i, k, a, b];
t[n_, m_] = If[m == 0, n!, Product[Sum[(-1)^(i + k)*StirlingS1[k - 1, i]*(m + 1)^i, {i, 0, k - 1}], {k, 1, n}]];
a = Table[Table[t[n, m], {n, 0, 10}], {m, 0, 10}];
b = Table[Table[a[[m, n - m + 1]], {m, n, 1, -1}], {n, 1, Length[a]}];
Flatten[%]

t(n,k)=If[m == 0, n!, Product[Sum[(-1)^(i + k)*StirlingS1[k - 1, i]*(m + 1)^i, {i, 0, k - 1}], {k, 1, n}]];

out_(n,k)=Antidiagonal(t(n,k)).

A335997 Triangle read by rows: T(n,k) = Product_{i=n-k+1..n} i! for 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 6, 12, 12, 1, 24, 144, 288, 288, 1, 120, 2880, 17280, 34560, 34560, 1, 720, 86400, 2073600, 12441600, 24883200, 24883200, 1, 5040, 3628800, 435456000, 10450944000, 62705664000, 125411328000, 125411328000

Offset: 0

Werner Schulte, Jul 08 2020

Based on some integer sequence a(n), n>0, define triangular arrays A(a;n,k) by recurrence: A(a;0,0) = 1, and A(a;i,j) = 0 if j<0 or j>i, and A(a;n,k) = n! / (n-k)! * A(a;n-1,k) + a(n) * A(a;n-1,k-1) for 0<=k<=n. Then, Product_{i=1..n} (1 + (a(i) / i!) * x) = Sum_{k=0..n} A(a;n,k) / T(n,k) * x^k for n>=0 with empty product 1 (case n=0).

For the row reversed triangle R(n,k) = Product_{i=k+1..n} i! with empty product 1 (case k=n) the terms of the matrix inverse M are given by M(n,n) = 1 for n >= 0 and M(n,n-1) = -n! for n > 0 otherwise 0. - Werner Schulte, Oct 25 2022

			The triangle starts:
n\k :  0     1      2        3         4         5         6
============================================================
  0 :  1
  1 :  1     1
  2 :  1     2      2
  3 :  1     6     12       12
  4 :  1    24    144      288       288
  5 :  1   120   2880    17280     34560     34560
  6 :  1   720  86400  2073600  12441600  24883200  24883200
  etc.

Cf. A000012 (col_0), A000142 (col_1), A010790 (col_2), A176037 (col_3), A000178 (main diagonal and first subdiagonal).
Row sums equal A051399(n+1).
Cf. A009963, A093002, A193520.

T[n_, k_] := Product[i!, {i, n - k + 1, n}]; Table[T[n, k], {n, 0, 7}, {k, 0, n}] // Flatten (* Amiram Eldar, Jul 08 2020 *)

T(n,k) = T(n,1) * T(n-1,k-1) for 0 < k <= n.
T(2*n,n) = A093002(n+1) for n >= 0.
T(n,k)/T(k,k) = A009963(n,k) for 0 <= k <= n.
(Sum_{k=0..n} T(n,k) * T(n,n-k))/T(n,n) = A193520(n) for n >= 0.

cp's OEIS Frontend

A156584 Triangle T(n,k) = SF(n+1)/(SF(n-k+1)*SF(k+1)) where SF(n) is the superfactorial A000178(n), read by rows.

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A193521 G.f.: A(x) = ( Sum_{n>=0} x^n/sf(n) )^3 where A(x) = Sum_{n>=0} a(n)*x^n/sf(n), and sf(n) = Product_{k=0..n} k! is the superfactorial of n (A000178).

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A156586 A new q-combination type general triangle sequence based on Stirling first polynomials: here q=4: m=3: t(n,k)=If[m == 0, n!, Product[Sum[(-1)^(i + k)*StirlingS1[k - 1, i]*(m + 1)^i, {i, 0, k - 1}], {k, 1, n}]]; b(n,k,m)=If[n == 0, 1, t[n, m]/(t[k, m]*t[n - k, m])].

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A156587 A new q-combination type general triangle sequence based on Stirling first polynomials: here q=5: m=4: t(n,k)=If[m == 0, n!, Product[Sum[(-1)^(i + k)*StirlingS1[k - 1, i]*(m + 1)^i, {i, 0, k - 1}], {k, 1, n}]]; b(n,k,m)=If[n == 0, 1, t[n, m]/(t[k, m]*t[n - k, m])].

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A156588 A triangle of q factorial type based on Stirling first polynomials: t(n,k)=If[m == 0, n!, Product[Sum[(-1)^(i + k)*StirlingS1[k - 1, i]*(m + 1)^i, {i, 0, k - 1}], {k, 1, n}]].

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A335997 Triangle read by rows: T(n,k) = Product_{i=n-k+1..n} i! for 0 <= k <= n.

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A156586 A new q-combination type general triangle sequence based on Stirling first polynomials: here q=4: m=3: t(n,k)=If[m == 0, n!, Product[Sum[(-1)^(i + k)StirlingS1[k - 1, i](m + 1)^i, {i, 0, k - 1}], {k, 1, n}]]; b(n,k,m)=If[n == 0, 1, t[n, m]/(t[k, m]*t[n - k, m])].

A156587 A new q-combination type general triangle sequence based on Stirling first polynomials: here q=5: m=4: t(n,k)=If[m == 0, n!, Product[Sum[(-1)^(i + k)StirlingS1[k - 1, i](m + 1)^i, {i, 0, k - 1}], {k, 1, n}]]; b(n,k,m)=If[n == 0, 1, t[n, m]/(t[k, m]*t[n - k, m])].

A156588 A triangle of q factorial type based on Stirling first polynomials: t(n,k)=If[m == 0, n!, Product[Sum[(-1)^(i + k)StirlingS1[k - 1, i](m + 1)^i, {i, 0, k - 1}], {k, 1, n}]].