A007696 -id:A007696 - cp's OEIS frontend

A049029 Triangle read by rows, the Bell transform of the quartic factorial numbers A007696(n+1) without column 0.

1, 5, 1, 45, 15, 1, 585, 255, 30, 1, 9945, 5175, 825, 50, 1, 208845, 123795, 24150, 2025, 75, 1, 5221125, 3427515, 775845, 80850, 4200, 105, 1, 151412625, 108046575, 27478710, 3363045, 219450, 7770, 140, 1, 4996616625, 3824996175, 1069801425

Offset: 1

Wolfdieter Lang

Previous name was: Triangle of numbers related to triangle A048882; generalization of Stirling numbers of second kind A008277, Lah-numbers A008297, ...
a(n,m) enumerates unordered n-vertex m-forests composed of m plane increasing quintic (5-ary) trees. Proof based on the a(n,m) recurrence. See also the F. Bergeron et al. reference, especially Table 1, first row and Example 1 for the e.g.f. for m=1. - Wolfdieter Lang, Sep 14 2007
Also the Bell transform of A007696(n+1). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 28 2016

			Triangle starts:
{1};
{5,1};
{45,15,1};
{585,255,30,1};
{9945,5175,825,50,1};
...

F. Bergeron, Ph. Flajolet and B. Salvy, Varieties of Increasing Trees, in Lecture Notes in Computer Science vol. 581, ed. J.-C. Raoult, Springer 1992, pp. 24-48.
P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, arXiv:quant-ph/0402027, 2004.
T. Copeland, Mathemagical Forests
T. Copeland, Addendum to Mathemagical Forests
T. Copeland, A Class of Differential Operators and the Stirling Numbers
M. Janjic, Some classes of numbers and derivatives, JIS 12 (2009) #09.8.3.
W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
W. Lang, First 10 rows.
Shi-Mei Ma, Some combinatorial sequences associated with context-free grammars, arXiv:1208.3104v2 [math.CO]. - From N. J. A. Sloane, Aug 21 2012
E. Neuwirth, Recursively defined combinatorial Functions: Extending Galton's board, Discr. Maths. 239 (2001) 33-51.
Mathias Pétréolle, Alan D. Sokal, Lattice paths and branched continued fractions. II. Multivariate Lah polynomials and Lah symmetric functions, arXiv:1907.02645 [math.CO], 2019.

a(n, m) := S2(5, n, m) is the fifth triangle of numbers in the sequence S2(1, n, m) := A008277(n, m) (Stirling 2nd kind), S2(2, n, m) := A008297(n, m) (Lah), S2(3, n, m) := A035342(n, m), S2(4, n, m) := A035469(n, m). a(n, 1)= A007696(n). A007559(n).
Cf. A048882, A007696. Row sums: A049120(n), n >= 1.
Cf. A094638

# The function BellMatrix is defined in A264428.
# Adds (1,0,0,0, ..) as column 0.
BellMatrix(n -> mul(4*k+1, k=0..n), 9); # Peter Luschny, Jan 28 2016

a[n_, m_] /; n >= m >= 1 := a[n, m] = (4(n-1) + m)*a[n-1, m] + a[n-1, m-1]; a[n_, m_] /; n < m = 0; a[, 0] = 0; a[1, 1] = 1; Flatten[Table[a[n, m], {n, 1, 9}, {m, 1, n}]] (* _Jean-François Alcover, Jul 22 2011 *)
rows = 9;
a[n_, m_] := BellY[n, m, Table[Product[4k+1, {k, 0, j}], {j, 0, rows}]];
Table[a[n, m], {n, 1, rows}, {m, 1, n}] // Flatten (* Jean-François Alcover, Jun 22 2018 *)

a(n, m) = n!*A048882(n, m)/(m!*4^(n-m)); a(n+1, m) = (4*n+m)*a(n, m)+ a(n, m-1), n >= m >= 1; a(n, m) := 0, n

a(n, m) = sum(|A051142(n, j)|*S2(j, m), j=m..n) (matrix product), with S2(j, m) := A008277(j, m) (Stirling2 triangle). Priv. comm. to W. Lang by E. Neuwirth, Feb 15 2001; see also the 2001 Neuwirth reference. See the general comment on products of Jabotinsky matrices given under A035342.

From Peter Bala, Nov 25 2011: (Start)

E.g.f.: G(x,t) = exp(t*A(x)) = 1+t*x+(5*t+t^2)*x^2/2!+(45*t+15*t^2+t^3)*x^3/3!+..., where A(x) = -1+(1-4*x)^(-1/4) satisfies the autonomous differential equation A'(x) = (1+A(x))^5.

The generating function G(x,t) satisfies the partial differential equation t*(dG/dt+G) = (1-4*x)*dG/dx, from which follows the recurrence given above.

The row polynomials are given by D^n(exp(x*t)) evaluated at x = 0, where D is the operator (1+x)^5*d/dx. Cf. A008277 (D = (1+x)*d/dx), A105278 (D = (1+x)^2*d/dx), A035342 (D = (1+x)^3*d/dx) and A035469 (D = (1+x)^4*d/dx).

(End)

New name from Peter Luschny, Jan 30 2016

A051617 a(n) = (4n+5)(!^4)/5(!^4), related to A007696(n+1) ((4n+1)(!^4) quartic, or 4-factorials).

Original entry on oeis.org

1, 9, 117, 1989, 41769, 1044225, 30282525, 999323325, 36974963025, 1515973484025, 68218806781125, 3342721532275125, 177164241210581625, 10098361749003152625, 616000066689192310125, 40040004334797500158125

Offset: 0

Wolfdieter Lang

Row m=5 of the array A(5; m,n) := ((4*n+m)(!^4))/m(!^4), m >= 0, n >= 0.

G. C. Greubel, Table of n, a(n) for n = 0..363

Cf. A047053, A007696(n+1), A000407, A034176(n+1), A034177(n+1) (rows m=0..4).

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(1/(1-4*x)^(9/4))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Aug 15 2018

s=1;lst={s};Do[s+=n*s;AppendTo[lst, s], {n, 8, 5!, 4}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 08 2008 *)
With[{nn = 30}, CoefficientList[Series[1/(1 - 4*x)^(9/4), {x, 0, nn}], x]*Range[0, nn]!] (* G. C. Greubel, Aug 15 2018 *)

x='x+O('x^30); Vec(serlaplace(1/(1-4*x)^(9/4))) \\ G. C. Greubel, Aug 15 2018

a(n) = ((4*n+5)(!^4))/5(!^4).

E.g.f.: 1/(1-4*x)^(9/4).

A265604 Triangle read by rows: The inverse Bell transform of the quartic factorial numbers (A007696).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, -2, 3, 1, 0, 10, -5, 6, 1, 0, -80, 30, -5, 10, 1, 0, 880, -290, 45, 5, 15, 1, 0, -12320, 3780, -560, 35, 35, 21, 1, 0, 209440, -61460, 8820, -735, 0, 98, 28, 1, 0, -4188800, 1192800, -167300, 14700, -735, 0, 210, 36, 1

Offset: 0

Peter Luschny, Dec 30 2015

			[ 1]
[ 0,      1]
[ 0,      1,      1]
[ 0,     -2,      3,      1]
[ 0,     10,     -5,      6,      1]
[ 0,    -80,     30,     -5,     10,      1]
[ 0,    880,   -290,     45,      5,     15,      1]

Peter Luschny, The Bell transform
Richell O. Celeste, Roberto B. Corcino, and Ken Joffaniel M. Gonzales. Two Approaches to Normal Order Coefficients, Journal of Integer Sequences, Vol. 20 (2017), Article 17.3.5.

Cf. A007696, A264428, A264429.

Inverse Bell transforms of other multifactorials are: A048993, A049404, A049410, A075497, A075499, A075498, A119275, A122848, A265605.

# uses[bell_transform from A264428]
def inverse_bell_matrix(generator, dim):
    G = [generator(k) for k in srange(dim)]
    row = lambda n: bell_transform(n, G)
    M = matrix(ZZ, [row(n)+[0]*(dim-n-1) for n in srange(dim)]).inverse()
    return matrix(ZZ, dim, lambda n,k: (-1)^(n-k)*M[n,k])
multifact_4_1 = lambda n: prod(4*k + 1 for k in (0..n-1))
print(inverse_bell_matrix(multifact_4_1, 8))

A034255 Related to quartic factorial numbers A007696.

Original entry on oeis.org

1, 10, 120, 1560, 21216, 297024, 4243200, 61526400, 902387200, 13355330560, 199115837440, 2986737561600, 45030812467200, 681895160217600, 10364806435307520, 158063298138439680, 2417438677411430400, 37067393053641932800, 569667303771760230400, 8772876478085107548160

Offset: 1

Wolfdieter Lang

Michael De Vlieger, Table of n, a(n) for n = 1..833
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seq., Vol. 3 (2000), Article 00.2.4.
Elżbieta Liszewska and Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.
Index entries for sequences related to factorial numbers.

First column of triangle A048882.

Cf. A007696, A068466.

Rest[CoefficientList[Series[(-1+(1-16x)^(-1/4))/4,{x,0,20}],x]] (* Harvey P. Dale, May 19 2011 *)

a(n) = 4^(n-1)*A007696(n)/n!, where A007696(n) = (4*n-3)(!^4) = Product_{j=1..n} (4*j-3), n >= 1.
G.f.: (-1+(1-16*x)^(-1/4))/4.
a(n) = A048882(n, 1).
Convolution of A034385(n-1) with A025749(n), n >= 1.
D-finite with recurrence: n*a(n) + 4*(-4*n+3)*a(n-1) = 0. - R. J. Mathar, Jan 28 2020
a(n) ~ 2^(4*n-2) * n^(-3/4) / Gamma(1/4). - Amiram Eldar, Aug 18 2025

A051621 a(n) = (4n+9)(!^4)/9(!^4), related to A007696(n+1) ((4n+1)(!^4) quartic, or 4-factorials).

Original entry on oeis.org

1, 13, 221, 4641, 116025, 3364725, 111035925, 4108329225, 168441498225, 7579867420125, 371413503586125, 19684915690064625, 1122040194333683625, 68444451854354701125, 4448889370533055573125, 306973366566780834545625

Offset: 0

Wolfdieter Lang

Row m=9 of the array A(5; m,n) := ((4*n+m)(!^4))/m(!^4), m >= 0, n >= 0.

G. C. Greubel, Table of n, a(n) for n = 0..363

Cf. A047053, A007696(n+1), A000407, A034176(n+1), A034177(n+1), A051617-A051622 (rows m=0..10).

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(1/(1-4*x)^(13/4))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Aug 15 2018

s=1;lst={s};Do[s+=n*s;AppendTo[lst, s], {n, 12, 5!, 4}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 08 2008 *)
With[{nn = 30}, CoefficientList[Series[1/(1 - 4*x)^(13/4), {x, 0, nn}], x]*Range[0, nn]!] (* G. C. Greubel, Aug 15 2018 *)

x='x+O('x^30); Vec(serlaplace(1/(1-4*x)^(13/4))) \\ G. C. Greubel, Aug 15 2018

a(n) = ((4*n+9)(!^4))/9(!^4) = A007696(n+3)/(5*9).

E.g.f.: 1/(1-4*x)^(13/4).

A265606 Triangle read by rows: The Bell transform of the quartic factorial numbers (A007696).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 5, 3, 1, 0, 45, 23, 6, 1, 0, 585, 275, 65, 10, 1, 0, 9945, 4435, 990, 145, 15, 1, 0, 208845, 89775, 19285, 2730, 280, 21, 1, 0, 5221125, 2183895, 456190, 62965, 6370, 490, 28, 1, 0, 151412625, 62002395, 12676265, 1715490, 171255, 13230, 798, 36, 1

Offset: 0

Peter Luschny, Dec 30 2015

			[1],
[0, 1],
[0, 1, 1],
[0, 5, 3, 1],
[0, 45, 23, 6, 1],
[0, 585, 275, 65, 10, 1],
[0, 9945, 4435, 990, 145, 15, 1],
[0, 208845, 89775, 19285, 2730, 280, 21, 1],

Richell O. Celeste, Roberto B. Corcino, and Ken Joffaniel M. Gonzales. Two Approaches to Normal Order Coefficients, Journal of Integer Sequences, Vol. 20 (2017), Article 17.3.5.
Peter Luschny, The Bell transform

Cf. A007696, A264428, A264429.

Bell transforms of other multifactorials are: A000369, A004747, A039683, A051141, A051142, A119274, A132062, A132393, A203412.

(* The function BellMatrix is defined in A264428. *)
rows = 10;
M = BellMatrix[Pochhammer[1/4, #] 4^# &, rows];
Table[M[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jul 23 2019 *)

# uses[bell_transform from A264428]
def A265606_row(n):
    multifact_4_1 = lambda n: prod(4*k + 1 for k in (0..n-1))
    mfact = [multifact_4_1(k) for k in (0..n)]
    return bell_transform(n, mfact)
[A265606_row(n) for n in (0..7)]

A169619 Hankel transform of quartic (or 4-fold) factorial numbers A007696.

Original entry on oeis.org

1, 4, 640, 11059200, 39749419008000, 48575443603606732800000, 29918051262318914110928977920000000, 12898757220773940360062849214160935321600000000000

Offset: 0

Paul Barry, Dec 03 2009

Cf. A007696.

Table[Product[((4*k+1)*(4*k+4))^(n-k), {k,0,n}], {n,0,10}] (* Vaclav Kotesovec, Jan 23 2024 *)

a(n) = Product_{k=0..n} ((4*k+1)*(4*k+4))^(n-k).

a(n) ~ 2^(2*n^2 + 3*n + 5/8) * Pi^(n + 5/8) * n^(n^2 + 5*n/4 + 35/96) / (A^(7/8) * Gamma(1/4)^(n + 1/4) * exp(3*n^2/2 + 5*n/4 - 7/96 - G/(4*Pi))), where G is Catalan's constant A006752 and A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Jan 23 2024

A048994 Triangle of Stirling numbers of first kind, s(n,k), n >= 0, 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, -1, 1, 0, 2, -3, 1, 0, -6, 11, -6, 1, 0, 24, -50, 35, -10, 1, 0, -120, 274, -225, 85, -15, 1, 0, 720, -1764, 1624, -735, 175, -21, 1, 0, -5040, 13068, -13132, 6769, -1960, 322, -28, 1, 0, 40320, -109584, 118124, -67284, 22449, -4536, 546, -36, 1, 0, -362880, 1026576, -1172700, 723680, -269325, 63273, -9450, 870, -45, 1

Offset: 0

N. J. A. Sloane

The unsigned numbers are also called Stirling cycle numbers: |s(n,k)| = number of permutations of n objects with exactly k cycles.
Mirror image of the triangle A054654. - Philippe Deléham, Dec 30 2006
Also the triangle gives coefficients T(n,k) of x^k in the expansion of C(x,n) = (a(k)*x^k)/n!. - Mokhtar Mohamed, Dec 04 2012
From Wolfdieter Lang, Nov 14 2018: (Start)
This is the Sheffer triangle of Jabotinsky type (1, log(1 + x)). See the e.g.f. of the triangle below.
This is the inverse Sheffer triangle of the Stirling2 Sheffer triangle A008275.
The a-sequence of this Sheffer triangle (see a W. Lang link in A006232)
is from the e.g.f. A(x) = x/(exp(x) -1) a(n) = Bernoulli(n) = A027641(n)/A027642(n), for n >= 0. The z-sequence vanishes.
The Boas-Buck sequence for the recurrences of columns has o.g.f. B(x) = Sum_{n>=0} b(n)*x^n = 1/((1 + x)*log(1 + x)) - 1/x. b(n) = (-1)^(n+1)*A002208(n+1)/A002209(n+1), b = {-1/2, 5/12, -3/8, 251/720, -95/288, 19087/60480,...}. For the Boas-Buck recurrence of Riordan and Sheffer triangles see the Aug 10 2017 remark in A046521, adapted to the Sheffer case, also for two references. See the recurrence and example below. (End)
Let G(n,m,k) be the number of simple labeled graphs on [n] with m edges and k components. Then T(n,k) = Sum (-1)^m*G(n,m,k). See the Read link below. Equivalently, T(n,k) = Sum mu(0,p) where the sum is over all set partitions p of [n] containing k blocks and mu is the Moebius function in the incidence algebra associated to the set partition lattice on [n]. - Geoffrey Critzer, May 11 2024

			Triangle begins:
  n\k 0     1       2       3      4      5      6    7    8   9 ...
  0   1
  1   0     1
  2   0    -1       1
  3   0     2      -3       1
  4   0    -6      11      -6      1
  5   0    24     -50      35    -10      1
  6   0  -120     274    -225     85    -15      1
  7   0   720   -1764    1624   -735    175    -21    1
  8   0 -5040   13068  -13132   6769  -1960    322  -28    1
  9   0 40320 -109584  118124 -67284  22449  -4536  546  -36   1
  ... - _Wolfdieter Lang_, Aug 22 2012
------------------------------------------------------------------
From _Wolfdieter Lang_, Nov 14 2018: (Start)
Recurrence: s(5,2)= s(4, 1) - 4*s(4, 2) = -6 - 4*11 = -50.
Recurrence from the a- and z-sequences: s(6, 3) = 2*(1*1*(-50) + 3*(-1/2)*35 + 6*(1/6)*(-10) + 10*0*1) = -225.
Boas-Buck recurrence for column k = 3, with b = {-1/2, 5/12, -3/8, ...}:
s(6, 3) = 6!*((-3/8)*1/3! + (5/12)*(-6)/4! + (-1/2)*35/5!) = -225. (End)

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 833.
L. Comtet, Advanced Combinatorics, Reidel, 1974; Chapter V, also p. 310.
J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, p. 93.
F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 226.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 245.
J. Riordan, An Introduction to Combinatorial Analysis, p. 48.

T. D. Noe, Rows n=0..100 of triangle, flattened
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Paul Barry, Generalized Stirling Numbers, Exponential Riordan Arrays, and Toda Chain Equations, Journal of Integer Sequences, 17 (2014), #14.2.3.
Fatima Zohra Bensaci, Rachid Boumahdi, and Laala Khaldi, Finite Sums Involving Fibonacci and Lucas Numbers, J. Int. Seq. (2024). See p. 9.
R. M. Dickau, Stirling numbers of the first kind. [Illustrates the unsigned Stirling cycle numbers A132393.]
Askar Dzhumadil'daev and Damir Yeliussizov, Walks, partitions, and normal ordering, Electronic Journal of Combinatorics, 22(4) (2015), #P4.10.
Bill Gosper, Colored illustrations of triangle of Stirling numbers of first kind read mod 2, 3, 4, 5, 6, 7.
Gergő Nemes, An Asymptotic Expansion for the Bernoulli Numbers of the Second Kind, J. Int. Seq. 14 (2011), #11.4.8.
A. Hennessy and Paul Barry, Generalized Stirling Numbers, Exponential Riordan Arrays, and Orthogonal Polynomials, J. Int. Seq. 14 (2011), #11.8.2 (A-number typo A048894).
NIST Digital Library of Mathematical Functions, Stirling Numbers
Ken Ono, Larry Rolen, and Florian Sprung, Zeta-Polynomials for modular form periods, p. 6, arXiv:1602.00752 [math.NT], 2016.
Ricardo A. Podestá, New identities for binary Krawtchouk polynomials, binomial coefficients and Catalan numbers, arXiv preprint arXiv:1603.09156 [math.CO], 2016.
Ronald Read, An Introduction to Chromatic Polynomials, Journal of Combinatorial Theory, 4(1968)52-71.
Wikipedia, Stirling numbers and exponential generating functions.

See especially A008275 which is the main entry for this triangle. A132393 is an unsigned version, and A008276 is another version.
Cf. A008277, A039814-A039817, A048993, A084938.
A000142(n) = Sum_{k=0..n} |s(n, k)| for n >= 0.
Row sums give A019590(n+1).
Cf. A002209, A027641/A027642, A216294, A271705.

a048994 n k = a048994_tabl !! n !! k
a048994_row n = a048994_tabl !! n
a048994_tabl = map fst $ iterate (\(row, i) ->
(zipWith (-) ([0] ++ row) $ map (* i) (row ++ [0]), i + 1)) ([1], 0)
-- Reinhard Zumkeller, Mar 18 2013

A048994:= proc(n,k) combinat[stirling1](n,k) end: # R. J. Mathar, Feb 23 2009
seq(print(seq(coeff(expand(k!*binomial(x,k)),x,i),i=0..k)),k=0..9); # Peter Luschny, Jul 13 2009
A048994_row := proc(n) local k; seq(coeff(expand(pochhammer(x-n+1,n)), x,k), k=0..n) end: # Peter Luschny, Dec 30 2010

Table[StirlingS1[n, m], {n, 0, 9}, {m, 0, n}] (* Peter Luschny, Dec 30 2010 *)

create_list(stirling1(n,k),n,0,12,k,0,n); /* Emanuele Munarini, Mar 11 2011 */

a(n,k) = if(k<0 || k>n,0, if(n==0,1,(n-1)*a(n-1,k)+a(n-1,k-1)))

trg(nn)=for (n=0, nn-1, for (k=0, n, print1(stirling(n,k,1), ", ");); print();); \\ Michel Marcus, Jan 19 2015

s(n, k) = A008275(n,k) for n >= 1, k = 1..n; column k = 0 is {1, repeat(0)}.
s(n, k) = s(n-1, k-1) - (n-1)*s(n-1, k), n, k >= 1; s(n, 0) = s(0, k) = 0; s(0, 0) = 1.
The unsigned numbers a(n, k)=|s(n, k)| satisfy a(n, k)=a(n-1, k-1)+(n-1)*a(n-1, k), n, k >= 1; a(n, 0) = a(0, k) = 0; a(0, 0) = 1.
Triangle (signed) = [0, -1, -1, -2, -2, -3, -3, -4, -4, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, ...]; Triangle(unsigned) = [0, 1, 1, 2, 2, 3, 3, 4, 4, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, ...]; where DELTA is Deléham's operator defined in A084938.
Sum_{k=0..n} (-m)^(n-k)*s(n, k) = A000142(n), A001147(n), A007559(n), A007696(n), ... for m = 1, 2, 3, 4, ... .- Philippe Deléham, Oct 29 2005
A008275*A007318 as infinite lower triangular matrices. - Gerald McGarvey, Aug 20 2009
T(n,k) = n!*[x^k]([t^n]exp(x*log(1+t))). - Peter Luschny, Dec 30 2010, updated Jun 07 2020
From Wolfdieter Lang, Nov 14 2018: (Start)
Recurrence from the Sheffer a-sequence (see a comment above): s(n, k) = (n/k)*Sum_{j=0..n-k} binomial(k-1+j, j)*Bernoulli(j)*s(n-1, k-1+j), for n >= 1 and k >= 1, with s(n, 0) = 0 if n >= 1, and s(0,0) = 1.
Boas-Buck type recurrence for column k: s(n, k) = (n!*k/(n - k))*Sum_{j=k..n-1} b(n-1-j)*s(j, k)/j!, for n >= 1 and k = 0..n-1, with input s(n, n) = 1. For sequence b see the Boas-Buck comment above. (End)
T(n,k) = Sum_{j=k..n} (-1)^(n-j)*A271705(n,j)*A216294(j,k). - Mélika Tebni, Feb 23 2023

Offset corrected by R. J. Mathar, Feb 23 2009

Formula corrected by Philippe Deléham, Sep 10 2009

A001813 Quadruple factorial numbers: a(n) = (2n)!/n!.

Original entry on oeis.org

1, 2, 12, 120, 1680, 30240, 665280, 17297280, 518918400, 17643225600, 670442572800, 28158588057600, 1295295050649600, 64764752532480000, 3497296636753920000, 202843204931727360000, 12576278705767096320000, 830034394580628357120000, 58102407620643984998400000

Offset: 0

N. J. A. Sloane

Counts binary rooted trees (with out-degree <= 2), embedded in plane, with n labeled end nodes of degree 1. Unlabeled version gives Catalan numbers A000108.
Define a "downgrade" to be the permutation which places the items of a permutation in descending order. We are concerned with permutations that are identical to their downgrades. Only permutations of order 4n and 4n+1 can have this property; the number of permutations of length 4n having this property are equinumerous with those of length 4n+1. If a permutation p has this property then the reversal of this permutation also has it. a(n) = number of permutations of length 4n and 4n+1 that are identical to their downgrades. - Eugene McDonnell (eemcd(AT)mac.com), Oct 26 2003
Number of broadcast schemes in the complete graph on n+1 vertices, K_{n+1}. - Calin D. Morosan (cd_moros(AT)alumni.concordia.ca), Nov 28 2008
Hankel transform is A137565. - Paul Barry, Nov 25 2009
The e.g.f. of 1/a(n) = n!/(2*n)! is (exp(sqrt(x)) + exp(-sqrt(x)) )/2. - Wolfdieter Lang, Jan 09 2012
From Tom Copeland, Nov 15 2014: (Start)
Aerated with intervening zeros (1,0,2,0,12,0,120,...) = a(n) (cf. A123023 and A001147), the e.g.f. is e^(t^2), so this is the base for the Appell sequence with e.g.f. e^(t^2) e^(x*t) = exp(P(.,x),t) (reverse A059344, cf. A099174, A066325 also). P(n,x) = (a. + x)^n with (a.)^n = a_n and comprise the umbral compositional inverses for e^(-t^2)e^(x*t) = exp(UP(.,x),t), i.e., UP(n,P(.,t)) = x^n = P(n,UP(.,t)), e.g., (P(.,t))^n = P(n,t).
Equals A000407*2 with leading 1 added. (End)
a(n) is also the number of square roots of any permutation in S_{4*n} whose disjoint cycle decomposition consists of 2*n transpositions. - Luis Manuel Rivera Martínez, Mar 04 2015
Self-convolution gives A076729. - Vladimir Reshetnikov, Oct 11 2016
For n > 1, it follows from the formula dated Aug 07 2013 that a(n) is a Zumkeller number (A083207). - Ivan N. Ianakiev, Feb 28 2017
For n divisible by 4, a(n/4) is the number of ways to place n points on an n X n grid with pairwise distinct abscissae, pairwise distinct ordinates, and 90-degree rotational symmetry. For n == 1 (mod 4), the number of ways is a((n-1)/4) because the center point can be considered "fixed". For 180-degree rotational symmetry see A006882, for mirror symmetry see A000085, A135401, and A297708. - Manfred Scheucher, Dec 29 2017

			The following permutations of order 8 and their reversals have this property:
  1 7 3 5 2 4 0 6
  1 7 4 2 5 3 0 6
  2 3 7 6 1 0 4 5
  2 4 7 1 6 0 3 5
  3 2 6 7 0 1 5 4
  3 5 1 7 0 6 2 4

D. E. Knuth, The Art of Computer Programming, Vol. 4, Section 7.2.1.6, Eq. 32.
L. C. Larson, The number of essentially different nonattacking rook arrangements, J. Recreat. Math., 7 (No. 3, 1974), circa pages 180-181.
Eugene McDonnell, "Magic Squares and Permutations" APL Quote-Quad 7.3 (Fall, 1976)
R. W. Robinson, Counting arrangements of bishops, pp. 198-214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976).
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, Table of n, a(n) for n = 0..100
Marcello Artioli, Giuseppe Dattoli, Silvia Licciardi, and Rosa Maria Pidatella, Hermite and Laguerre Functions: a Unifying Point of View, Università degli Studi di Catania, Sicily, Italy (2019).
Murray Bremner and Martin Markl, Distributive laws between the Three Graces, arXiv:1809.08191 [math.AT], 2018.
R. B. Brent, Generalizing Tuenter's Binomial Sums, J. Int. Seq. 18 (2015) # 15.3.2.
Peter J. Cameron, Some treelike objects, Quart. J. Math. Oxford Ser. (2) 38 (1987), no. 150, 155--183. MR0891613 (89a:05009). See p. 155. - _N. J. A. Sloane_, Apr 18 2014
Peter J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Elliot J. Carr and Matthew J. Simpson, New homogenization approaches for stochastic transport through heterogeneous media, arXiv:1810.08890 [physics.bio-ph], 2018.
W. Y. C. Chen, L. W. Shapiro and L. L. M. Young, Parity reversing involutions on plane trees and 2-Motzkin paths, arXiv:math/0503300 [math.CO], 2005.
Ali Chouria, Vlad-Florin Drǎgoi, and Jean-Gabriel Luque, On recursively defined combinatorial classes and labelled trees, arXiv:2004.04203 [math.CO], 2020.
P. Cvitanovic, Group theory for Feynman diagrams in non-Abelian gauge theories, Phys. Rev. D14 (1976), 1536-1553.
Nick Early, Honeycomb tessellations and canonical bases for permutohedral blades, arXiv:1810.03246 [math.CO], 2018.
John Engbers, David Galvin, and Clifford Smyth, Restricted Stirling and Lah numbers and their inverses, arXiv:1610.05803 [math.CO], 2016. See p. 8.
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 127
S. Goodenough and C. Lavault, On subsets of Riordan subgroups and Heisenberg--Weyl algebra, arXiv preprint arXiv:1404.1894 [cs.DM], 2014.
S. Goodenough and C. Lavault, Overview on Heisenberg—Weyl Algebra and Subsets of Riordan Subgroups, The Electronic Journal of Combinatorics, 22(4) (2015), #P4.16,
H. W. Gould, A class of binomial sums and a series transform, Utilitas Math., 45 (1994), 71-83. (Annotated scanned copy)
A. M. Ibrahim, Extension of factorial concept to negative numbers, Notes on Number Theory and Discrete Mathematics, Vol. 19, 2013, 2, 30-42.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 115
L. C. Larson, The number of essentially different nonattacking rook arrangements, J. Recreat. Math., 7 (No. 3, 1974), circa pages 180-181. [Annotated scan of pages 180 and 181 only]
Jesús Leaños, Rutilo Moreno, and Luis Manuel Rivera-Martínez, On the number of mth roots of permutations, Australas. J. Combin. 52 (2012), 41-54 (Theorem 1).
Jesús Leaños, Rutilo Moreno, and Luis Manuel Rivera-Martínez, On the number of mth roots of permutations, arXiv:1005.1531 [math.CO], 2010-2011.
Édouard Lucas, Théorie des Nombres, Gauthier-Villars, Paris, 1891, Vol. 1, p. 221.
R. J. Marsh and P. P. Martin, Pascal arrays: counting Catalan sets, arXiv:math/0612572 [math.CO], 2006.
Calin D. Morosan, On the number of broadcast schemes in networks, Information Processing Letters, Volume 100, Issue 5 (2006), 188-193.
R. A. Proctor, Let's Expand Rota's Twelvefold Way for Counting Partitions!, arXiv:math/0606404 [math.CO], 2006-2007.
C. Radoux, Déterminants de Hankel et théorème de Sylvester, Séminaire Lotharingien de Combinatoire, B28b (1992), 9 pp.
J. Riordan, Letter to N. J. A. Sloane, Feb 03 1975 (with notes by njas)
H. E. Salzer, Coefficients for expressing the first thirty powers in terms of the Hermite polynomials, Math. Comp., 3 (1948), 167-169.
Elena L. Wang and Guoce Xin, On Ward Numbers and Increasing Schröder Trees, arXiv:2507.15654 [math.CO], 2025. See pp. 12-13.
Index to divisibility sequences
Index entries for related partition-counting sequences

Cf. A037224, A048854, A001147, A007696, A008545, A122670 (essentially the same sequence), A000165, A047055, A047657, A084947, A084948, A084949, A010050, A000142, A008275, A000108, A000984, A008276, A000680, A094216.
Catalan(n-1)*M^(n-1)*n! for M=1,2,3,4,5,6: A001813, A052714 (or A144828), A221954, A052734, A221953, A221955.
Cf. A123023, A059344, A099174, A066325, A001700, A000407, A006882.
Cf. A000085, A135401, A297708. - Manfred Scheucher, Jan 07 2018

List([0..20],n->Factorial(2*n)/Factorial(n)); # Muniru A Asiru, Nov 01 2018

[Factorial(2*n)/Factorial(n): n in [0..20]]; // Vincenzo Librandi, Oct 09 2018

A001813 := n->(2*n)!/n!;
A001813 := n -> mul(k, k = select(k-> k mod 4 = 2,[$1 .. 4*n])):
seq(A001813(n), n=0..16);  # Peter Luschny, Jun 23 2011

Table[(2n)!/n!, {n,0,20}] (* Harvey P. Dale, May 02 2011 *)

makelist(binomial(n+n, n)*n!,n,0,30); /* Martin Ettl, Nov 05 2012 */

a(n)=binomial(n+n,n)*n! \\ Charles R Greathouse IV, Jun 15 2011

first(n) = x='x+O('x^n); Vec(serlaplace((1 - 4*x)^(-1/2))) \\ Iain Fox, Jan 01 2018 (corrected by Iain Fox, Jan 11 2018)

from math import factorial
def A001813(n): return factorial(n<<1)//factorial(n) # Chai Wah Wu, Feb 14 2023

[binomial(2*n,n)*factorial(n) for n in range(0, 17)] # Zerinvary Lajos, Dec 03 2009

E.g.f.: (1-4*x)^(-1/2).
a(n) = (2*n)!/n! = Product_{k=0..n-1} (4*k + 2) = A081125(2*n).
Integral representation as n-th moment of a positive function on a positive half-axis: a(n) = Integral_{x=0..oo} x^n*exp(-x/4)/(sqrt(x)*2*sqrt(Pi)) dx, n >= 0. This representation is unique. - Karol A. Penson, Sep 18 2001
Define a'(1)=1, a'(n) = Sum_{k=1..n-1} a'(n-k)*a'(k)*C(n, k); then a(n)=a'(n+1). - Benoit Cloitre, Apr 27 2003
With interpolated zeros (1, 0, 2, 0, 12, ...) this has e.g.f. exp(x^2). - Paul Barry, May 09 2003
a(n) = A000680(n)/A000142(n)*A000079(n) = Product_{i=0..n-1} (4*i + 2) = 4^n*Pochhammer(1/2, n) = 4^n*GAMMA(n+1/2)/sqrt(Pi). - Daniel Dockery (peritus(AT)gmail.com), Jun 13 2003
For asymptotics, see the Robinson paper.
a(k) = (2*k)!/k! = Sum_{i=1..k+1} |A008275(i,k+1)| * k^(i-1). - André F. Labossière, Jun 21 2007
a(n) = 12*A051618(a) n >= 2. - Zerinvary Lajos, Feb 15 2008
a(n) = A000984(n)*A000142(n). - Zerinvary Lajos, Mar 25 2008
a(n) = A016825(n-1)*a(n-1). - Roger L. Bagula, Sep 17 2008
a(n) = (-1)^n*A097388(n). - D. Morosan (cd_moros(AT)alumni.concordia.ca), Nov 28 2008
From Paul Barry, Jan 15 2009: (Start)
G.f.: 1/(1-2x/(1-4x/(1-6x/(1-8x/(1-10x/(1-... (continued fraction);
a(n) = (n+1)!*A000108(n). (End)
a(n) = Sum_{k=0..n} A132393(n,k)*2^(2n-k). - Philippe Deléham, Feb 10 2009
G.f.: 1/(1-2x-8x^2/(1-10x-48x^2/(1-18x-120x^2/(1-26x-224x^2/(1-34x-360x^2/(1-42x-528x^2/(1-... (continued fraction). - Paul Barry, Nov 25 2009
a(n) = A173333(2*n,n) for n>0; cf. A006963, A001761. - Reinhard Zumkeller, Feb 19 2010
From Gary W. Adamson, Jul 19 2011: (Start)
a(n) = upper left term of M^n, M = an infinite square production matrix as follows:
  2, 2, 0, 0, 0, 0, ...
  4, 4, 4, 0, 0, 0, ...
  6, 6, 6, 6, 0, 0, ...
  8, 8, 8, 8, 8, 0, ...
  ...
(End)
a(n) = (-2)^n*Sum_{k=0..n} 2^k*s(n+1,n+1-k), where s(n,k) are the Stirling numbers of the first kind, A048994. - Mircea Merca, May 03 2012
G.f.: 1/Q(0), where Q(k) = 1 + x*(4*k+2) - x*(4*k+4)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 18 2013
G.f.: 2/G(0), where G(k) = 1 + 1/(1 - x*(8*k+4)/(x*(8*k+4) - 1 + 8*x*(k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 30 2013
G.f.: G(0)/2, where G(k) = 1 + 1/(1 - 2*x/(2*x + 1/(2*k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 01 2013
D-finite with recurrence: a(n) = (4*n-6)*a(n-2) + (4*n-3)*a(n-1), n>=2. - Ivan N. Ianakiev, Aug 07 2013
Sum_{n>=0} 1/a(n) = (exp(1/4)*sqrt(Pi)*erf(1/2) + 2)/2 = 1 + A214869, where erf(x) is the error function. - Ilya Gutkovskiy, Nov 10 2016
Sum_{n>=0} (-1)^n/a(n) = 1 - sqrt(Pi)*erfi(1/2)/(2*exp(1/4)), where erfi(x) is the imaginary error function. - Amiram Eldar, Feb 20 2021
a(n) = 1/([x^n] hypergeom([1], [1/2], x/4)). - Peter Luschny, Sep 13 2024
a(n) = 2^n*n!*JacobiP(n, -1/2, -n, 3). - Peter Luschny, Jan 22 2025
G.f.: 2F0(1,1/2;;4x). - R. J. Mathar, Jun 07 2025

More terms from James Sellers, May 01 2000

A007559 Triple factorial numbers (3*n-2)!!! with leading 1 added.

Original entry on oeis.org

1, 1, 4, 28, 280, 3640, 58240, 1106560, 24344320, 608608000, 17041024000, 528271744000, 17961239296000, 664565853952000, 26582634158080000, 1143053268797440000, 52580450364682240000, 2576442067869429760000, 133974987529210347520000, 7368624314106569113600000

Offset: 0

N. J. A. Sloane

a(n) is the number of increasing quaternary trees on n vertices. (See A001147 for ternary and A000142 for binary trees.) - David Callan, Mar 30 2007
a(n) is the product of the positive integers k <= 3*n that have k modulo 3 = 1. - Peter Luschny, Jun 23 2011
See A094638 for connections to differential operators. - Tom Copeland, Sep 20 2011
Partial products of A016777. - Reinhard Zumkeller, Sep 20 2013
For n > 2, a(n) is a Zumkeller number. - Ivan N. Ianakiev, Jan 28 2020
a(n) is the number of generalized permutations of length n related to the degenerate Eulerian numbers (see arXiv:2007.13205), cf. A336633. - Orli Herscovici, Jul 28 2020

			G.f. = 1 + x + 4*x^2 + 28*x^3 + 280*x^4 + 3640*x^5 + 58240*x^6 + ...
a(3) = 28 and a(4) = 280; with top row of M^3 = (28, 117, 108, 27), sum = 280.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n=0..100
Alexander Burstein and Louis W. Shapiro, Pseudo-involutions in the Riordan group, arXiv:2112.11595 [math.CO], 2021.
P. Codara, O. M. D'Antona and P. Hell, A simple combinatorial interpretation of certain generalized Bell and Stirling numbers, arXiv preprint arXiv:1308.1700 [cs.DM], 2013.
S. Goodenough and C. Lavault, On subsets of Riordan subgroups and Heisenberg--Weyl algebra, arXiv preprint arXiv:1404.1894 [cs.DM], 2014.
S. Goodenough and C. Lavault, Overview on Heisenberg—Weyl Algebra and Subsets of Riordan Subgroups, The Electronic Journal of Combinatorics, 22(4) (2015), #P4.16.
Orli Herscovici, Generalized permutations related to the degenerate Eulerian numbers, arXiv preprint arXiv:2007.13205 [math.CO], 2020.
Ivan N. Ianakiev, A simple proof that for n > 2, a(n) is a Zumkeller number
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
J.-C. Novelli and J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962 [math.CO], 2014.
M. D. Schmidt, Generalized j-Factorial Functions, Polynomials, and Applications, J. Int. Seq. 13 (2010), 10.6.7, Table 6.3.

Cf. A001147, A004987, A008544, A032031, A051141, A129116.
a(n)= A035469(n, 1), n >= 1, (first column of triangle A035469(n, m)).
Cf. A107716. - Gary W. Adamson, Oct 22 2009
Cf. A095660. - Gary W. Adamson, Jul 19 2011
Subsequence of A007661. A007696, A008548.
a(n) = A286718(n,0), n >= 0.
Row sums of A336633.

List([0..20], n-> Product([0..n-1], k-> 3*k+1 )); # G. C. Greubel, Aug 20 2019

a007559 n = a007559_list !! n
a007559_list = scanl (*) 1 a016777_list
-- Reinhard Zumkeller, Sep 20 2013

b:= func< n | (n lt 2) select n else (3*n-2)*Self(n-1) >;
[1] cat [b(n): n in [1..20]]; // G. C. Greubel, Aug 20 2019

A007559 := n -> mul(k, k = select(k-> k mod 3 = 1, [$1 .. 3*n])): seq(A007559(n), n = 0 .. 17); # Peter Luschny, Jun 23 2011
# second Maple program:
b:= proc(n) option remember; `if`(n<1, 1, n*b(n-3)) end:
a:= n-> b(3*n-2):
seq(a(n), n=0..20);  # Alois P. Heinz, Dec 18 2024

a[ n_] := If[ n < 0, 1 / Product[ k, {k, - 2, 3 n - 1, -3}],
  Product[ k, {k, 1, 3 n - 2, 3}]]; (* Michael Somos, Oct 14 2011 *)
FoldList[Times,1,Range[1,100,3]] (* Harvey P. Dale, Jul 05 2013 *)
Range[0, 19]! CoefficientList[Series[((1 - 3 x)^(-1/3)), {x, 0, 19}], x] (* Vincenzo Librandi, Oct 08 2015 *)

a(n):=if n=1 then 1 else (n)!*(sum(m/n*sum(binomial(k,n-m-k)*(-1/3)^(n-m-k)* binomial (k+n-1,n-1),k,1,n-m),m,1,n)+1); /* Vladimir Kruchinin, Aug 09 2010 */

{a(n) = if( n<0, (-1)^n / prod(k=0,-1-n, 3*k + 2), prod(k=0, n-1, 3*k + 1))}; /* Michael Somos, Oct 14 2011 */

my(x='x+O('x^33)); Vec(serlaplace((1-3*x)^(-1/3))) /* Joerg Arndt, Apr 24 2011 */

def A007559(n) : return mul(j for j in range(1,3*n,3))
[A007559(n) for n in (0..17)]  # Peter Luschny, May 20 2013

a(n) = Product_{k=0..n-1} (3*k + 1).
a(n) = (3*n - 2)!!!.
a(n) = A007661(3*n-2).
E.g.f.: (1-3*x)^(-1/3).
a(n) ~ sqrt(2*Pi)/Gamma(1/3)*n^(-1/6)*(3*n/e)^n*(1 - (1/36)/n - ...). - Joe Keane (jgk(AT)jgk.org), Nov 22 2001
a(n) = 3^n*Pochhammer(1/3, n).
a(n) = Sum_{k=0..n} (-3)^(n-k)*A048994(n, k). - Philippe Deléham, Oct 29 2005
a(n) = n!*(1+Sum_{m=1..n} (m/n)*Sum_{k=1..n-m} binomial(k, n-m-k)*(-1/3)^(n-m-k)*binomial(k+n-1, n-1)), n>1. - Vladimir Kruchinin, Aug 09 2010
From Gary W. Adamson, Jul 19 2011: (Start)
a(n) = upper left term in M^n, M = a variant of Pascal (1,3) triangle (Cf. A095660); as an infinite square production matrix:
  1, 3, 0, 0, 0,...
  1, 4, 3, 0, 0,...
  1, 5, 7, 3, 0,...
  ...
  a(n+1) = sum of top row terms of M^n. (End)
a(n) = (-2)^n*Sum_{k=0..n} (3/2)^k*s(n+1,n+1-k), where s(n,k) are the Stirling numbers of the first kind, A048994. - Mircea Merca, May 03 2012
G.f.: 1/Q(0) where Q(k) = 1 - x*(3*k+1)/( 1 - x*(3*k+3)/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Mar 21 2013
G.f.: G(0)/2, where G(k)= 1 + 1/(1 - x*(3*k+1)/(x*(3*k+1) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 26 2013
E.g.f.: E(0)/2, where E(k)= 1 + 1/(1 - x*(3*k+1)/(x*(3*k+1) + (k+1)/E(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 26 2013
Let D(x) = 1/sqrt(1 - 2*x) be the e.g.f. for the sequence of double factorial numbers A001147. Then the e.g.f. A(x) for the triple factorial numbers satisfies D( Integral_{t=0..x} A(t) dt ) = A(x). Cf. A007696 and A008548. - Peter Bala, Jan 02 2015
O.g.f.: hypergeom([1, 1/3], [], 3*x). - Peter Luschny, Oct 08 2015
a(n) = 3^n * Gamma(n + 1/3)/Gamma(1/3). - Artur Jasinski, Aug 23 2016
a(n) = sigma[3,1]^{(n)}n, n >= 0, with the elementary symmetric function of degree n in the n numbers 1, 4, 7, ..., 1+3*(n-1), with sigma[3,1]^{(n)}_0 := 1. See the first formula. - _Wolfdieter Lang, May 29 2017
a(n) = (-1)^n / A008544(n), 0 = a(n)*(+3*a(n+1) -a(n+2)) +a(n+1)*a(n+1) for all n in Z. - Michael Somos, Sep 30 2018
D-finite with recurrence: a(n) +(-3*n+2)*a(n-1)=0, n>=1. - R. J. Mathar, Feb 14 2020
Sum_{n>=1} 1/a(n) = (e/9)^(1/3) * (Gamma(1/3) - Gamma(1/3, 1/3)). - Amiram Eldar, Jun 29 2020

Better description from Wolfdieter Lang

Showing 1-10 of 84 results. Next

cp's OEIS Frontend

A049029 Triangle read by rows, the Bell transform of the quartic factorial numbers A007696(n+1) without column 0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A051617 a(n) = (4*n+5)(!^4)/5(!^4), related to A007696(n+1) ((4*n+1)(!^4) quartic, or 4-factorials).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A265604 Triangle read by rows: The inverse Bell transform of the quartic factorial numbers (A007696).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Sage

A034255 Related to quartic factorial numbers A007696.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A051621 a(n) = (4*n+9)(!^4)/9(!^4), related to A007696(n+1) ((4*n+1)(!^4) quartic, or 4-factorials).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A265606 Triangle read by rows: The Bell transform of the quartic factorial numbers (A007696).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Sage

A169619 Hankel transform of quartic (or 4-fold) factorial numbers A007696.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A048994 Triangle of Stirling numbers of first kind, s(n,k), n >= 0, 0 <= k <= n.

Views

Author

Keywords

Comments

Examples

A051617 a(n) = (4n+5)(!^4)/5(!^4), related to A007696(n+1) ((4n+1)(!^4) quartic, or 4-factorials).

A051621 a(n) = (4n+9)(!^4)/9(!^4), related to A007696(n+1) ((4n+1)(!^4) quartic, or 4-factorials).