A051295 -id:A051295 - cp's OEIS frontend

A143965 Factorial eigentriangle: A119502 * (A051295 *0^(n-k)); 0 <= k <= n.

1, 1, 1, 2, 1, 2, 6, 2, 2, 5, 24, 6, 4, 5, 15, 120, 24, 12, 10, 15, 54, 720, 120, 48, 30, 30, 54, 235, 5040, 720, 240, 120, 90, 108, 235, 1237, 40320, 5040, 1440, 600, 360, 324, 470, 1237, 7790

Offset: 0

Gary W. Adamson, Sep 06 2008

Triangle read by rows, termwise product of (n-k)! (i.e factorial decrescendo,
A119502) and the INVERT transform of the factorials (A051295) prefaced by a 1:
(1, 1, 2, 5, 15, 54, 235, 1237, 7790, ...). A119502 = (1; 1,1; 2,1,1; 6,2,1,1; 24,6,2,1,1; ...).
The operation (A051295 * 0^(n-k)) with A051295 prefaced with a 1 = an infinite lower triangular matrix with (1, 1, 2, 5, 15, 54, 235, ...) in the main diagonal and the rest zeros.
Row sums = the INVERT transform of the factorials, A051295: (1, 2, 5, 15, 54, 235, 1237, ...).
Right border shifts A051295: (1, 1, 2, 5, 15, ...).
Sum of n-th row terms = rightmost term of next row; e.g. ( 6 + 2 + 2 + 5) = 15.
With offset 1 for n and k, T(n,k) counts permutations of [n] that contain a 132 pattern only as part of a 4132 pattern by position k of largest entry n. Example: T(5,3)=4 counts 34512, 34521, 43512, 43521. - David Callan, Nov 21 2011
From Gary W. Adamson, Jul 21 2016: (Start)
A production matrix M for the reversal of the triangle is follows: M =
  1, 1, 0, 0, 0, 0, ...
  1, 0, 2, 0, 0, 0, ...
  1, 0, 0, 3, 0, 0, ...
  1, 0, 0, 0, 4, 0, ...
  1, 0, 0, 0, 0, 5, ...
... Take powers of M, extracting the top row, getting: (1), (1, 1), (2, 1, 2), (5, 2, 2, 6), ... (End)

			First few rows of the triangle:
     1;
     1,   1;
     2,   1,   2;
     6,   2,   2,   5;
    24,   6,   4,   5, 15;
   120,  24,  12,  10, 15,  54;
   720, 120,  48,  30, 30,  54, 235;
  5040, 720, 240, 120, 90, 108, 235, 1737;
  ...
Example: Row 3 = (6, 2, 2, 5) = termwise products of row 3 terms of triangle A119502 (6, 2, 1, 1) and the first four terms of (1, 1, 2, 5, ...) = (6*1, 2*1, 1*2, 1*5).

Cf. A000142, A051295, A119502.

Factorial eigentriangle: A119502 * (A051295 *0^(n-k)); 0 <= k <= n.

The operation uses A119502 prefaced with a 1 = (1, 1, 2, 5, 15, 54, 235, ...); i.e., the right border of the triangle.

A084938 Triangle read by rows: T(n,k) = Sum_{j>=0} j!*T(n-j-1, k-1) for n >= 0, k >= 0.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 6, 5, 3, 1, 0, 24, 16, 9, 4, 1, 0, 120, 64, 31, 14, 5, 1, 0, 720, 312, 126, 52, 20, 6, 1, 0, 5040, 1812, 606, 217, 80, 27, 7, 1, 0, 40320, 12288, 3428, 1040, 345, 116, 35, 8, 1, 0, 362880, 95616, 22572, 5768, 1661, 519, 161, 44, 9, 1

Offset: 0

Philippe Deléham, Jul 16 2003; corrections Dec 17 2008, Dec 20 2008, Feb 05 2009

Triangle T(n,k) is [0,1,1,2,2,3,3,4,4,...] DELTA [1,0,0,0,0,0,...] = A110654 DELTA A000007.
In general, the triangle [r_0,r_1,r_2,r_3,...] DELTA [s_0,s_1,s_2,s_3,...] has generating function 1/(1-(r_0*x+s_0*x*y)/(1-(r_1*x+s_1*x*y)/(1-(r_2*x+s_2*x*y)/(1-(r_3*x+s_3*x*y)/(1-...(continued fraction). See also the Formula section below.
T(n,k) = number of permutations on [n] that (i) contain a 132 pattern only as part of a 4132 pattern and (ii) start with n+1-k. For example, for n >= 1, T(n,1) = (n-1)! counts all (n-1)! permutations on [n] that start with n: either they avoid 132 altogether or the initial entry serves as the "4" in a 4132 pattern and T(4,3) = 3 counts 2134, 2314, 2341. - David Callan, Jul 20 2005
T(n,k) is the number of permutations on [n] that (i) contain a (scattered) 342 pattern only as part of a 1342 pattern and (ii) contain 1 in position k. For example, T(4,3) counts 3214, 4213, 4312. (It does not count, say, 2314 because 231 forms an offending 342 pattern.) - David Callan, Jul 20 2005
Riordan array (1,x*g(x)) where g(x) is the g.f. of the factorials (n!). - Paul Barry, Sep 25 2008
Modulo 2, this sequence becomes A106344.
T(n,k) is the number of permutations of {1,2,...,n} having k cycles such that the elements of each cycle of the permutation form an interval. - Ran Pan, Nov 11 2016
The convolution triangle of the factorial numbers. - Peter Luschny, Oct 09 2022

			From _Paul Barry_, Sep 25 2008: (Start)
Triangle [0,1,1,2,2,3,3,4,4,5,5,...] DELTA [1,0,0,0,0,...] begins
  1;
  0,      1;
  0,      1,     1;
  0,      2,     2,     1;
  0,      6,     5,     3,    1;
  0,     24,    16,     9,    4,    1;
  0,    120,    64,    31,   14,    5,   1;
  0,    720,   312,   126,   52,   20,   6,   1;
  0,   5040,  1812,   606,  217,   80,  27,   7,  1;
  0,  40320, 12288,  3428, 1040,  345, 116,  35,  8, 1;
  0, 362880, 95616, 22572, 5768, 1661, 519, 161, 44, 9, 1. (End)
From _Paul Barry_, May 14 2009: (Start)
The production matrix is
  0,   1;
  0,   1,  1;
  0,   1,  1, 1;
  0,   2,  1, 1, 1;
  0,   7,  2, 1, 1, 1;
  0,  34,  7, 2, 1, 1, 1;
  0, 206, 34, 7, 2, 1, 1, 1;
which is based on A075834. (End)

T. D. Noe, Rows n = 0..100 of triangle, flattened
Paul Barry, A Note on a One-Parameter Family of Catalan-Like Numbers, JIS 12 (2009) 09.5.4.
Paul Barry, Continued fractions and transformations of integer sequences, JIS 12 (2009), Article 09.7.6.
Paul Barry, On the inversion of Riordan arrays, arXiv:2101.06713 [math.CO], 2021.
Paul Barry and A. Hennessy, A Note on Narayana Triangles and Related Polynomials, Riordan Arrays, and MIMO Capacity Calculations, J. Int. Seq. 14 (2011), Article 11.3.8.
David Callan, A combinatorial interpretation of the eigensequence for composition, arXiv:math/0507169 [math.CO], 2005.
David Callan, A Combinatorial Interpretation of the Eigensequence for Composition, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.4.
H. Fuks and J. M. G. Soto, Exponential convergence to equilibrium in cellular automata asymptotically emulating identity, arXiv preprint arXiv:1306.1189 [nlin.CG], 2013.
Sergey Kitaev and Philip B. Zhang, Distributions of mesh patterns of short lengths, arXiv:1811.07679 [math.CO], 2018.
Peter Luschny, Transformations of integer sequences.
R. J. Mathar, Properties of Deleham's Delta Transformation: OEIS A084938

Cf. A003149, A051295 (row sums), A052186, A090238,
Cf. A287899.
Columns: A000007, A000142, A003149, A090595, A090319.
Diagonals: A000012, A001477, A000096, A092286, A090386, A090391, A090392, A090393, A090394.

function T(n,k) // T = A084938
  if k lt 0 or k gt n then return 0;
  elif n eq 0 or k eq n then return 1;
  elif k eq 0 then return 0;
  else return (&+[Factorial(j)*T(n-j-1,k-1): j in [0..n-1]]);
  end if; return T;
end function;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 10 2022

DELTA := proc(r,s,n) local T,x,y,q,P,i,j,k,t1; T := array(0..n,0..n);
for i from 0 to n do q[i] := x*r[i+1]+y*s[i+1]; od: for k from 0 to n do P[0,k] := 1; od: for i from 0 to n do P[i,-1] := 0; od:
for i from 1 to n do for k from 0 to n do P[i,k] := sort(expand(P[i,k-1] + q[k]*P[i-1,k+1])); od: od:
for i from 0 to n do t1 := P[i,0]; for j from 0 to i do T[i,j] := coeff(coeff(t1,x,i-j),y,j); od: lprint( seq(T[i,j],j=0..i) ); od: end;
# To produce the current triangle: s3 := n->floor((n+1)/2); s4 := n->if n = 0 then 1 else 0; fi; r := [seq(s3(i),i= 0..40)]; s := [seq(s4(i),i=0..40)]; DELTA(r,s,20);
# Uses function PMatrix from A357368.
PMatrix(10, n -> factorial(n - 1)); # Peter Luschny, Oct 09 2022

a[0, 0] = 1; a[n_, k_] := a[n, k] = Sum[j! a[n - j - 1, k - 1], {j, 0, n - 1}]; Flatten[Table[a[i, j], {i, 0, 10}, {j, 0, i}]] (* T. D. Noe, Feb 22 2012 *)
DELTA[r_, s_, m_] := Module[{p, q, t, x, y}, q[k_] := x*r[[k+1]] + y*s[[k+1]]; p[0, ] = 1; p[, -1] = 0; p[n_ /; n >= 1, k_ /; k >= 0] := p[n, k] = p[n, k-1] + q[k]*p[n-1, k+1] // Expand; t[n_, k_] := Coefficient[p[n, 0], x^(n-k)*y^k]; t[0, 0] = p[0, 0]; Table[t[n, k], {n, 0, m}, {k, 0, n}]]; DELTA[Floor[Range[10]/2], Prepend[Table[0, {10}], 1], 10] (* Jean-François Alcover, Sep 12 2013, after Philippe Deléham *)

def delehamdelta(R, S) :
    L = min(len(R), len(S)) + 1
    ring = PolynomialRing(ZZ, 'x')
    x = ring.gen()
    A = [Rk + x*Sk for Rk, Sk in zip(R, S)]
    C = [ring(0)] + [ring(1) for i in range(L)]
    for k in (1..L) :
        for n in range(k-1,0,-1) :
            C[n] = C[n-1] + C[n+1]*A[n-1]
        yield list(C[1])
def A084938_triangle(n) :
    for row in delehamdelta([(i+1)//2 for i in (0..n)], [0^i for i in (0..n)]):
        print(row)
A084938_triangle(10) # Peter Luschny, Jan 28 2012

The operator DELTA takes two sequences r = (r_0, r_1, ...), s = (s_0, s_1, ...) and produces a triangle T(n, k), 0 <= k <= n, as follows:
Let q(k) = x*r_k + y*s_k for k >= 0; let P(n, k) (n >= 0, k >= -1) be defined recursively by P(0, k) = 1 for k >= 0; P(n, -1) = 0 for n >= 1; P(n, k) = P(n, k-1) + q(k)*P(n-1, k+1) for n >= 1, k >= 0. Then P(n, k) is a homogeneous polynomial in x and y of degree n and T(n, k) = coefficient of x^(n-k)*y^k in P(n, 0).
T(n, n) = 1.
T(k+1, k) = A001477(k).
T(k+2, k) = A000096(k).
T(n+1, 1) = A000142(n).
T(n+2, 2) = A003149(n).
T(n+3, 3) = A090595(n).
T(n+4, 4) = A090319(n).
T(m+n, m) = Sum_{k=0..n} A090238(n, k)*binomial(m, k).
G.f. for column k: Sum_{n>=0} T(k+n, k)*x^n = (Sum_{n>=0} n!*x^n )^k.
For k>0, T(n+k, k) = Sum_{a_1 + a_2 + .. + a_k = n} (a_1)!*(a_2)!*..*(a_k)!; a_i>=0, n>=0.
T(n,k) = Sum_{j>=0} A075834(j)*T(n-1,k+j-1).
T(2n,n) = A287899(n). - Alois P. Heinz, Jun 02 2017
From G. C. Greubel, Nov 10 2022: (Start)
Sum_{k=0..n} T(n, k) = A051295(n).
Sum_{k=0..n} (-1)^k*T(n, k) = [n=0] - A052186(n-1)*[n>0]. (End)

Name edited by Derek Orr, May 01 2015

A055209 a(n) = Product_{i=0..n} i!^2.

Original entry on oeis.org

1, 1, 4, 144, 82944, 1194393600, 619173642240000, 15728001190723584000000, 25569049282962188245401600000000, 3366980847587422591723894776791040000000000, 44337041641882947649156022595410930014617600000000000000

Offset: 0

N. J. A. Sloane, Jul 18 2000

a(n) is the discriminant of the polynomial x(x+1)(x+2)...(x+n). - Yuval Dekel (dekelyuval(AT)hotmail.com), Nov 13 2003
This is the Hankel transform (see A001906 for definition) of the sequence: 1, 0, 1, 0, 5, 0, 61, 0, 1385, 0, 50521, ... (see A000364: Euler numbers). - Philippe Deléham, Apr 06 2005
Also, for n>0, the quotient of (-1)^(n-1)S(u)^(n^2)/S(un) and the determinant of the (n-1) X (n-1) square matrix [P'(u), P''(u), ..., P^(n-1)(u); P''(u), P'''(u), ..., P^(n)(u); P'''(u), P^(4)(u), ..., P^(n+1)(u); ...; P^(n-1)(u), P^(n)(u), ..., P^(2n-3)(u)] where S and P are the Weierstrass Sigma and The Weierstrass P-function, respectively and f^(n) is the n-th derivative of f. See the King and Schwarz & Weierstrass references. - Balarka Sen, Jul 31 2013
a(n) is the number of idempotent monotonic labeled magmas. That is, prod(i,j) >= max(i,j) and prod(i,i) = i. - Chad Brewbaker, Nov 03 2013
Ramanujan's infinite nested radical sqrt(1+2*sqrt(1+3*sqrt(1+...))) = 3 can be written sqrt(1+sqrt(4+sqrt(144+...))) = sqrt(a(1)+sqrt(a(2)+sqrt(a(3)+...))). Vijayaraghavan used that to prove convergence of Ramanujan's formula. - Petros Hadjicostas and Jonathan Sondow, Mar 22 2014
a(n) is the determinant of the (n+1)-th order Hankel matrix whose (i,j)-entry is equal to A000142(i+j), i,j = 0,1,...,n. - Michael Shmoish, Sep 02 2020

R. Bruce King, Beyond The Quartic Equation, Birkhauser Boston, Berlin, 1996, p. 72.
Srinivasa Ramanujan, J. Indian Math. Soc., III (1911), 90 and IV (1912), 226.
T. Vijayaraghavan, in Collected Papers of Srinivasa Ramanujan, G.H. Hardy, P.V. Seshu Aiyar and B.M. Wilson, eds., Cambridge Univ. Press, 1927, p. 348; reprinted by Chelsea, 1962.

G. C. Greubel, Table of n, a(n) for n = 0..32
Paul Barry, A Note on Three Families of Orthogonal Polynomials defined by Circular Functions, and Their Moment Sequences, Journal of Integer Sequences, Vol. 15 (2012), #12.7.2.
Richard Ehrenborg, The Hankel determinant of exponential polynomials, Amer. Math. Monthly, Vol. 107, No. 6 (2000), pp. 557-560.
William Q. Erickson and Jan Kretschmann, The structure and normalized volume of Monge polytopes, arXiv:2311.07522 [math.CO], 2023. See p. 7.
John W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5.
Christian Radoux, Déterminants de Hankel et théorème de Sylvester, Séminaire Lotharingien de Combinatoire, B28b (1992), 9 pp.
H. A. Schwarz and K. Weierstrass, Formeln und Lehrsätze zum Gebrauche der elliptischen Functionen, Springer, Berlin, 1893, p. 19.
Jonathan Sondow and Petros Hadjicostas, The generalized-Euler-constant function gamma(z) and a generalization of Somos's quadratic recurrence constant, J. Math. Anal. Appl., Vol. 332, No. 1 (2007), pp. 292-314; see pp. 305-306.
Michael Z. Spivey and Laura L. Steil, The k-Binomial Transforms and the Hankel Transform, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1.
Index entries for sequences related to factorial numbers

Cf. A055209 is the Hankel transform (see A001906 for definition) of A000023, A000142, A000166, A000522, A003701, A010842, A010843, A051295, A052186, A053486, A053487.

Cf. A112302, A258619.

[1] cat [(&*[(Factorial(k))^2: k in [1..n]]): n in [1..10]]; // G. C. Greubel, Oct 14 2018

seq(mul(mul(j^2,j=1..k), k=0..n), n=0..10); # Zerinvary Lajos, Sep 21 2007

Table[Product[(i!)^2,{i,n}],{n,0,11}] (* Harvey P. Dale, Jul 06 2011 *)
Table[BarnesG[n + 2]^2, {n, 0, 11}] (* Jan Mangaldan, May 07 2014 *)

a(n)=prod(i=1,n,i!)^2 \\ Charles R Greathouse IV, Jan 12 2012

def A055209(n) :
   return prod(factorial(i)^(2) for i in (0..n))
[A055209(n) for n in (0..11)] # Jani Melik, Jun 06 2015

a(n) = A000178(n)^2. - Philippe Deléham, Mar 06 2004
a(n) = Product_{i=0..n} i^(2*n - 2*i + 2). - Charles R Greathouse IV, Jan 12 2012
Asymptotic: a(n) ~ exp(2*zeta'(-1)-3/2*(1+n^2)-3*n)*(2*Pi)^(n+1)*(n+1)^ (n^2+2*n+5/6). - Peter Luschny, Jun 23 2012
lim_{n->infinity} a(n)^(2^(-(n+1))) = 1. - Vaclav Kotesovec, Jun 06 2015
Sum_{n>=0} 1/a(n) = A258619. - Amiram Eldar, Nov 17 2020

A051296 INVERT transform of factorial numbers.

Original entry on oeis.org

1, 1, 3, 11, 47, 231, 1303, 8431, 62391, 524495, 4960775, 52223775, 605595319, 7664578639, 105046841127, 1548880173119, 24434511267863, 410503693136559, 7315133279097607, 137787834979031839, 2734998201208351479, 57053644562104430735, 1247772806059088954855

Offset: 0

Leroy Quet

a(n) = Sum[ a1!a2!...ak! ] where (a1,a2,...,ak) ranges over all compositions of n. a(n) = number of trees on [0,n] rooted at 0, consisting entirely of filaments and such that the non-root labels on each filament, when arranged in order, form an interval of integers. A filament is a maximal path (directed away from the root) whose interior vertices all have outdegree 1 and which terminates at a leaf. For example with n=3, a(n) = 11 counts all n^(n-2) = 16 trees on [0,3] except the 3 trees {0->1, 1->2, 1->3}, {0->2, 2->1, 2->3}, {0->3, 3->1, 3->2} (they fail the all-filaments test) and the 2 trees {0->2, 0->3, 3->1}, {0->2, 0->1, 1->3} (they fail the interval-of-integers test). - David Callan, Oct 24 2004
a(n) is the number of lists of "unlabeled" permutations whose total length is n. "Unlabeled" means each permutation is on an initial segment of the positive integers (cf. A090238). Example: with dashes separating permutations, a(3) = 11 counts 123, 132, 213, 231, 312, 321, 1-12, 1-21, 12-1, 21-1, 1-1-1. - David Callan, Sep 20 2007
Number of compositions of n where there are k! sorts of part k. - Joerg Arndt, Aug 04 2014

			a(4) = 47 = 1*24 + 1*6 + 3*2 + 11*1.
a(4) = 47, the upper left term of M^4.

L. Comtet, Advanced Combinatorics, Reidel, 1974.

Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..440 (first 200 terms from Alois P. Heinz)
Jean-Paul Bultel, Ali Chouria, Jean-Gabriel Luque, and Olivier Mallet, Word symmetric functions and the Redfield-Polya theorem, hal-00793788, 2013.
Louis Comtet, Sur les coefficients de l'inverse de la série formelle Sum n! t^n, Comptes Rend. Acad. Sci. Paris, A 275 (1972), 569-572.
Richard Ehrenborg, Gábor Hetyei, and Margaret Readdy, Catalan-Spitzer permutations, arXiv:2310.06288 [math.CO], 2023. See p. 20.
Richard J. Martin, and Michael J. Kearney, Integral representation of certain combinatorial recurrences, Combinatorica: 35:3 (2015), 309-315.
Jun Yan, Results on pattern avoidance in parking functions, arXiv:2404.07958 [math.CO], 2024. See p. 7.

Cf. A051295, row sums of A090238.

Cf. A000142, A292778.

a:= proc(n) option remember; `if`(n<1, 1,
      add(a(n-i)*factorial(i), i=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Jul 28 2015

CoefficientList[Series[Sum[Sum[k!*x^k, {k, 1, 20}]^n, {n, 0, 20}], {x, 0, 20}], x] (* Geoffrey Critzer, Mar 22 2009 *)

h = lambda x: 1/(1-x*hypergeometric((1, 2), (), x))
taylor(h(x),x,0,22).list() # Peter Luschny, Jul 28 2015

def A051296_list(len):
    R, C = [1], [1]+[0]*(len-1)
    for n in (1..len-1):
        for k in range(n, 0, -1):
            C[k] = C[k-1] * k
        C[0] = sum(C[k] for k in (1..n))
        R.append(C[0])
    return R
print(A051296_list(23)) # Peter Luschny, Feb 21 2016

G.f.: 1/(1-Sum_{n>=1} n!*x^n).
a(0) = 1; a(n) = Sum_{k=1..n} a(n-k)*k! for n>0.
a(n) = Sum_{k>=0} A090238(n, k). - Philippe Deléham, Feb 05 2004
From Gary W. Adamson, Sep 26 2011: (Start)
a(n) is the upper left term of M^n, M = an infinite square production matrix as follows:
1, 1, 0, 0, 0, 0, ...
2, 0, 2, 0, 0, 0, ...
3, 0, 0, 3, 0, 0, ...
4, 0, 0, 0, 4, 0, ...
5, 0, 0, 0, 0, 5, ...
... (End)
G.f.: 1 + x/(G(0) - 2*x) where G(k) = 1 + (k+1)*x - x*(k+2)/G(k+1); (recursively defined continued fraction). - Sergei N. Gladkovskii, Dec 26 2012
a(n) ~ n! * (1 + 2/n + 7/n^2 + 35/n^3 + 216/n^4 + 1575/n^5 + 13243/n^6 + 126508/n^7 + 1359437/n^8 + 16312915/n^9 + 217277446/n^10), for coefficients see A260530. - Vaclav Kotesovec, Jul 28 2015
From Peter Bala, May 26 2017: (Start)
G.f. as an S-fraction: A(x) = 1/(1 - x/(1 - 2*x/(1 - x/(1 - 3*x/(1 - 2*x/(1 - 4*x/(1 - 3*x/(1 - n*x/(1 - (n - 1)*x/(1 - ...)))))))))). Cf. S-fraction for the o.g.f. of A000142.
A(x) = 1/(1 - x/(1 - x - x/(1 - 2*x/(1 - 2*x/(1 - 3*x/(1 - 3*x/(1 - 4*x/(1 - 4*x/(1 - ... ))))))))). (End)

Entry revised by David Callan, Sep 20 2007

A111146 Triangle T(n,k), read by rows, given by [0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, ...] DELTA [1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, ...] where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 0, 1, 0, 0, 2, 0, 0, 1, 4, 0, 0, 2, 5, 8, 0, 0, 6, 15, 17, 16, 0, 0, 24, 62, 68, 49, 32, 0, 0, 120, 322, 359, 243, 129, 64, 0, 0, 720, 2004, 2308, 1553, 756, 321, 128, 0, 0, 5040, 14508, 17332, 11903, 5622, 2151, 769, 256, 0, 0, 40320, 119664

Offset: 0

Philippe Deléham, Oct 19 2005

Let R(m,n,k), 0<=k<=n, the Riordan array (1, x*g(x)) where g(x) is g.f. of the m-fold factorials . Then Sum_{k, 0<=k<=n} = R(m,n,k) = Sum_{k, 0<=k<=n} T(n,k)*m^(n-k).
For m = -1, R(-1,n,k) is A026729(n,k).
For m = 0, R(0,n,k) is A097805(n,k).
For m = 1, R(1,n,k) is A084938(n,k).
For m = 2, R(2,n,k) is A111106(n,k).

			Triangle begins:
.1;
.0, 1;
.0, 0, 2;
.0, 0, 1, 4;
.0, 0, 2, 5, 8;
.0, 0, 6, 15, 17, 16;
.0, 0, 24, 62, 68, 49, 32;
.0, 0, 120, 322, 359, 243, 129, 64;
.0, 0, 720, 2004, 2308, 1553, 756, 321, 128;
.0, 0, 5040, 14508, 17332, 11903, 5622, 2151, 769, 256;
.0, 0, 40320, 119664, 148232, 105048, 49840, 18066, 5756, 1793, 512;
....................................................................
At y=2: Sum_{k=0..n} 2^k*T(n,k) = A113327(n) where (1 + 2*x + 8*x^2 + 36*x^3 +...+ A113327(n)*x^n +..) = 1/(1 - 2/1!*x*(1! + 2!*x + 3!*x^2 + 4!*x^3 +..) ).
At y=3: Sum_{k=0..n} 3^k*T(n,k) = A113328(n) where (1 + 3*x + 18*x^2 + 117*x^3 +...+ A113328(n)*x^n +..) = 1/(1 - 3/2!*x*(2! + 3!*x + 4!*x^2 + 5!*x^3 +..) ).
At y=4: Sum_{k=0..n} 4^k*T(n,k) = A113329(n) where (1 + 4*x + 32*x^2 + 272*x^3 +...+ A113329(n)*x^n +..) = 1/(1 - 4/3!*x*(3! + 4!*x + 5!*x^2 + 6!*x^3 +..) ).

Alois P. Heinz, Rows n = 0..140, flattened

Cf. A000045, A026729, A051295, A084938, A097805, A111106, A112934.
Cf. m-fold factorials : A000142, A001147, A007559, A007696, A008548, A008542.
Cf. A113326, A113327 (y=2), A113328 (y=3), A113329 (y=4), A113330 (y=5), A113331 (y=6).

T[n_, k_] := Module[{x = X + X*O[X]^n, y = Y + Y*O[Y]^k}, A = 1/(1 - x*y*Sum[x^j*Product[y + i, {i, 0, j - 1}], {j, 0, n}]); Coefficient[ Coefficient[A, X, n], Y, k]];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 26 2019, from PARI *)

{T(n,k)=local(x=X+X*O(X^n),y=Y+Y*O(Y^k)); A=1/(1-x*y*sum(j=0,n,x^j*prod(i=0,j-1,y+i))); return(polcoeff(polcoeff(A,n,X),k,Y))} (Hanna)

Sum_{k, 0<=k<=n} (-1)^(n-k)*T(n, k) = A000045(n+1), Fibonacci numbers.
Sum_{k, 0<=k<=n} T(n, k) = A051295(n).
Sum_{k, 0<=k<=n} 2^(n-k)*T(n, k) = A112934(n).
T(0, 0) = 1, T(n, n) = 2^(n-1).
G.f.: A(x, y) = 1/(1 - x*y*Sum_{j>=0} (y-1+j)!/(y-1)!*x^j ). - Paul D. Hanna, Oct 26 2005

A113143 Table T(n,k), n >= 0 and k >= 0, read by antidiagonals, related to A111146.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 2, 4, 1, 1, 2, 5, 8, 1, 1, 2, 6, 15, 16, 1, 1, 2, 7, 26, 54, 32, 1, 1, 2, 8, 41, 158, 235, 64, 1, 1, 2, 9, 60, 364, 1282, 1237, 128, 1, 1, 2, 10, 83, 708, 4409, 13158, 7790, 256, 1, 1, 2, 11, 110, 1226, 11428, 67563, 163354

Offset: 0

Philippe Deléham and Paul D. Hanna, Oct 30 2005

Let R(m,n,k), 0 <= k <= n, the Riordan array (1, x*g(x)) where g(x)is g.f. of the m-fold factorials.
Then the row sums of R(m,n,k) are given by row m; example: m = 1, R(1,n,k) = A084938(n,k) and A051295 gives the row sums of A084938.
Square array of INVERT of m-fold factorials.

			Table begins:
1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, ...
1, 1, 2, 5, 15, 54, 235, 1237, 7790, 57581, 489231, 4690254, ...
1, 1, 2, 6, 26, 158, 1282, 13158, 163354, 2374078, 39456386, ...
1, 1, 2, 7, 41, 364, 4409, 67573, 1248626, 26948347, 664414997, ...
1, 1, 2, 8, 60, 708, 11428, 232756, 5704964, 163192820, 5331728964, ...
1, 1, 2, 9, 83, 1226, 24727, 627909, 19169758, 682800001, 27776711627, ...
1, 1, 2, 10, 110, 1954, 47270, 1437562, 52531310, 2239259266, 109021857446, ...
1, 1, 2, 11, 141, 2928, 82597, 2925973, 124502114, 6179425823, 350316271761, ...
1, 1, 2, 12, 176, 4184, 134824, 5451528, 264710536, 14992543432, 969925065992, ...
1, 1, 2, 13, 215, 5758, 208643, 9481141, 517310894, 32922122485, 2393313188039, ...
1, 1, 2, 14, 258, 7686, 309322, 15604654, 945111938, 66766075046, 5387893860042, ...

Cf. A051295 (row n=1), A112934 (row n=2), A113144 (row n = 3), A113145 (row n=4), A113146 (row n=5), A113147 (row n = 6), A113148 (row n=7), A113149 (row n=8).

{T(n,k)=local(x=X+X*O(X^k),y=Y+Y*O(Y^k));A=1/(1-x*y*sum(j=0,k,x^j*prod(i= 0,j-1,y+i)));return(sum(m=0,k,n^(k-m)*polcoeff(polcoeff(A,k,X),m,Y)))}

T(n, k) = Sum_{j=0..k} n^(k-j)*A111146(k, j).

A113326 Table T(n,k), n>=1 and k>=0, read by antidiagonals, related to A111146.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 8, 5, 1, 4, 18, 36, 15, 1, 5, 32, 117, 176, 54, 1, 6, 50, 272, 801, 928, 235, 1, 7, 72, 525, 2400, 5724, 5296, 1237, 1, 8, 98, 900, 5675, 21792, 42633, 33024, 7790, 1, 9, 128, 1421, 11520, 62650, 203008, 331911, 227776, 57581

Offset: 0

Philippe Deléham and Paul D. Hanna, Oct 28 2005

			Table begins:
1,1,2,5,15,54,235,1237,7790,57581, 489231, ...
1,2,8,36,176,928,5296,33024,227776,1757504, ...
1,3,18,117,801,5724,42633,331911,2717874,23620329, ...
1,4,32,272,2400,21792,203008,1940224,19065344,193410560, ...
1,5,50,525,5675,62650,703975,8042625,93454750,1106250125, ...
1,6,72,900,11520,149904,1976400,26363232,355648320, ...
1,7,98,1421,21021,315168,4774021,72945859,1123559906, ...
1,8,128,2112,35456,601984,10306048,177639936,3080264704, ...
1,9,162,2997,56295,1067742,20392803,391614669,7555447854, ...
1,10,200,4100,85200,1785600,37644400,797224000,16946456000,
...

Cf. A111146, A051295 (row n=1), A113327 (row n=2), A113328 (row n=3), A113329 (row n=4), A113330 (row n=5), A113331 (row n=6).

{T(n,k)=local(y=Y+Y*O(Y^k)); polcoeff(1/(1-n/(n-1)!*y*sum(j=0,k,(n-1+j)!*y^j)),k,Y)}

T(n, k) = Sum_{j=0..k} n^j*A111146(k, j).

G.f. for row n: Sum_{k>=0}T(n, k)*y^k = 1/(1-n/(n-1)!*y*Sum_{j>=0}(n-1+j)!*y^j), for n>=1.

A381529 T(n,k) is the number of permutations of [n] having exactly k pairs of integers i=0, 0<=k<=A125811(n)-1, read by rows.

Original entry on oeis.org

1, 1, 2, 5, 1, 15, 5, 4, 54, 21, 24, 16, 5, 235, 89, 118, 112, 101, 35, 28, 2, 1237, 408, 577, 633, 719, 585, 402, 239, 167, 59, 14, 7790, 2106, 3023, 3529, 4410, 4463, 4600, 3012, 2789, 1933, 1438, 629, 442, 122, 34, 57581, 12529, 17693, 20980, 27208, 30064, 35359, 33332, 28137, 24970, 22850, 17148, 14272, 8645, 5639, 3684, 1809, 664, 282, 34

Offset: 0

Alois P. Heinz, Feb 26 2025

			T(4,0) = 15: (1)(2)(3)(4), (1,2)(3)(4), (1)(2,3)(4), (1)(2)(3,4), (1,2)(3,4), (1,2,3)(4), (1,3,2)(4), (1)(2,3,4), (1)(2,4,3), (1,2,3,4), (1,2,4,3), (1,3,2,4), (1,3,4,2), (1,4,2,3), (1,4,3,2).
T(4,1) = 5: (1)(2,4)(3), (1,2,4)(3), (1,4,2)(3), (1,3)(2)(4), (1,3)(2,4).
T(4,2) = 4: (1,4)(2)(3), (1,4)(2,3), (1,3,4)(2), (1,4,3)(2).
Triangle T(n,k) begins:
     1;
     1;
     2;
     5,   1;
    15,   5,   4;
    54,  21,  24,  16,   5;
   235,  89, 118, 112, 101,  35,  28,   2;
  1237, 408, 577, 633, 719, 585, 402, 239, 167, 59, 14;
  ...

Alois P. Heinz, Rows n = 0..55, flattened
Wikipedia, Inversion
Wikipedia, Permutation

Columns k=0-1 give: A051295, A381539.
Row sums give A000142.
Row lengths give A125811.
Last elements of rows give A381531.
Main diagonal gives A381545.
Cf. A008302, A125810 (similar for set partitions), A126673, A381299 (similar for ordered set partitions).

b:= proc(o, u, t) option remember; expand(`if`(u+o=0, max(0, t-1)!,
     `if`(t>0, b(u+o, 0$2)*(t-1)!, 0)+add(x^(u+j-1)*
        b(o-j, u+j-1, t+1), j=`if`(t=0, 1, 1..o))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0$2)):
seq(T(n), n=0..10);

Sum_{k>=1} k * T(n,k) = A126673(n)/2.

cp's OEIS Frontend

A260532 Coefficients in asymptotic expansion of sequence A051295.

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A143965 Factorial eigentriangle: A119502 * (A051295 *0^(n-k)); 0 <= k <= n.

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A084938 Triangle read by rows: T(n,k) = Sum_{j>=0} j!*T(n-j-1, k-1) for n >= 0, k >= 0.

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A055209 a(n) = Product_{i=0..n} i!^2.

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A051296 INVERT transform of factorial numbers.

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A111146 Triangle T(n,k), read by rows, given by [0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, ...] DELTA [1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, ...] where DELTA is the operator defined in A084938.

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A113143 Table T(n,k), n >= 0 and k >= 0, read by antidiagonals, related to A111146.

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