A000407 -id:A000407 - cp's OEIS frontend

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

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).

A295182 a(n) = n! * [x^n] exp(-n*x)/(1 - x)^n.

Original entry on oeis.org

1, 0, 2, 6, 72, 620, 8640, 122346, 2156672, 41367672, 905126400, 21646532270, 570077595648, 16268377195044, 502096929431552, 16629319748711250, 588938142209310720, 22196966267762213744, 887352465220427317248, 37496112562144553167062, 1670071417348195942400000, 78195398849926292810318940

Offset: 0

Ilya Gutkovskiy, Nov 16 2017

nonn

The n-th term of the n-fold exponential convolution of A000166 with themselves.

Robert Israel, Table of n, a(n) for n = 0..396
N. J. A. Sloane, Transforms
Index entries for sequences related to factorial numbers

Main diagonal of A295181.

Cf. A000166, A000407, A001813, A087981, A137775, A295183, A383379.

S:= series((exp(-x)/(1-x))^n,x,30):
seq(n!*coeff(S,x,n),n=0..29); # Robert Israel, Nov 16 2017

Table[n! SeriesCoefficient[Exp[-n x]/(1 - x)^n, {x, 0, n}], {n, 0, 21}]

a(n) = A295181(n,n).
a(n) ~ phi^(3*n - 1/2) * n^n / (5^(1/4) * exp(n*(1 + 1/phi))), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Nov 16 2017
a(n) = n! * Sum_{k=0..n} (-n)^(n-k) * binomial(n+k-1,k)/(n-k)!. - Seiichi Manyama, Apr 25 2025

A370915 A(n, k) = 4^n*Pochhammer(k/4, n). Square array read by ascending antidiagonals.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 5, 2, 1, 0, 45, 12, 3, 1, 0, 585, 120, 21, 4, 1, 0, 9945, 1680, 231, 32, 5, 1, 0, 208845, 30240, 3465, 384, 45, 6, 1, 0, 5221125, 665280, 65835, 6144, 585, 60, 7, 1, 0, 151412625, 17297280, 1514205, 122880, 9945, 840, 77, 8, 1

Offset: 0

Peter Luschny, Mar 06 2024

The sequence of square arrays A(m, n, k) starts: A094587 (m = 1), A370419 (m = 2), A371077(m = 3), this array (m = 4).

			The array starts:
[0] 1,    1,     1,     1,      1,      1,      1,      1,      1, ...
[1] 0,    1,     2,     3,      4,      5,      6,      7,      8, ...
[2] 0,    5,    12,    21,     32,     45,     60,     77,     96, ...
[3] 0,   45,   120,   231,    384,    585,    840,   1155,   1536, ...
[4] 0,  585,  1680,  3465,   6144,   9945,  15120,  21945,  30720, ...
[5] 0, 9945, 30240, 65835, 122880, 208845, 332640, 504735, 737280, ...
.
Seen as the triangle T(n, k) = A(n - k, k):
[0] 1;
[1] 0,      1;
[2] 0,      1,     1;
[3] 0,      5,     2,    1;
[4] 0,     45,    12,    3,   1;
[5] 0,    585,   120,   21,   4,  1;
[6] 0,   9945,  1680,  231,  32,  5, 1;
[7] 0, 208845, 30240, 3465, 384, 45, 6, 1;

Similar square arrays: A094587, A370419, A371077.
Rows: A000012, A001477, A028347, A370914.
Columns: A000007, A007696, A001813, A008545, A047053, A007696, A000407, A034176, A052570 and A034177, A051617, A051618, A051619, A051620.
Cf. A370913 (row sums of triangle), A371026.

A := (n, k) -> 4^n*pochhammer(k/4, n):
for n from 0 to 5 do seq(A(n, k), k = 0..9) od;
T := (n, k) -> A(n - k, k): seq(seq(T(n, k), k = 0..n), n = 0..9);
# Using the exponential generating functions of the columns:
EGFcol := proc(k, len) local egf, ser, n; egf := (1 - 4*x)^(-k/4);
ser := series(egf, x, len+2): seq(n!*coeff(ser, x, n), n = 0..len) end:
seq(lprint(EGFcol(n, 9)), n = 0..5);
# Using the generating polynomials for the rows:
P := (n, x) -> local k; add(Stirling1(n, k)*(-4)^(n - k)*x^k, k=0..n):
seq(lprint([n], seq(P(n, k), k = 0..8)), n = 0..5);
# Implementing the LU decomposition of A:
with(LinearAlgebra):
L := Matrix(7, 7, (n, k) -> A371026(n-1, k-1)):
U := Matrix(7, 7, (n, k) -> binomial(n-1, k-1)):
MatrixMatrixMultiply(L, Transpose(U));

A[n_, k_] := 4^n * Pochhammer[k/4, n]; Table[A[n - k, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, Mar 06 2024 *)

def A(n, k): return 4**n * rising_factorial(k/4, n)
for n in range(6): print([A(n, k) for k in range(9)])

A(n, k) = 4^n*Product_{j=0..n-1} (j + k/4).
A(n, k) = 4^n*Gamma(k/4 + n) / Gamma(k/4) for k >= 1.
The exponential generating function for column k is (1 - 4*x)^(-k/4). But much more is true: (1 - m*x)^(-k/m) are the exponential generating functions for the columns of the arrays A(m, n, k) = m^n*Pochhammer(k/m, n).
The polynomials P(n, x) = Sum_{k=0..n} Stirling1(n, k)*(-4)^(n-k)*x^k are ordinary generating functions for row n, i.e., A(n, k) = P(n, k).
In A370419 Werner Schulte pointed out how A371025 is related to the LU decomposition of A370419. Here the same procedure can be used and amounts to A = A371026 * transpose(binomial triangle), where '*' denotes matrix multiplication. See the Maple section for an implementation.

A001762 Number of labeled n-vertex dissections of a ball.

Original entry on oeis.org

1, 1, 10, 180, 4620, 152880, 6168960, 293025600, 15990004800, 984647664000, 67493121696000, 5094263446272000, 419688934689024000, 37465564582397952000, 3601861863990534144000, 370962724717928318976000, 40744403224500159055872000

Offset: 3

N. J. A. Sloane

nonn

This is the number of labeled Apollonian networks (planar 3-trees). - Allan Bickle, Feb 20 2024

			There is one maximal planar graph with 4 vertices, and one way to label it, so a(4) = 1.

L. W. Beineke and R. E. Pippert, Enumerating labeled k-dimensional trees and ball dissections, pp. 12-26 of Proceedings of Second Chapel Hill Conference on Combinatorial Mathematics and Its Applications, University of North Carolina, Chapel Hill, 1970. Reprinted in Math. Annalen, 191 (1971), 87-98.
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).

T. D. Noe, Table of n, a(n) for n = 3..100
L. W. Beineke and R. E. Pippert, The Number of Labeled Dissections of a k-Ball, Math. Annalen, 191 (1971), 87-98.
Allan Bickle, A Survey of Maximal k-degenerate Graphs and k-Trees, Theory and Applications of Graphs 0 1 (2024) Article 5.

Cf. A000407, A001763, A001764, A027610.

using Combinatorics
a(n) = n < 4 ? 1 : binomial(BigInt(n),3)*factorial(BigInt(3*n-9))÷factorial(BigInt(2*n-4))
print([a(n) for n in 3:28]) # Paul Muljadi, Mar 27 2024

Join[{1}, Table[Binomial[n, 3]*(3*n - 9)!/(2*n - 4)!, {n, 4, 25}]] (* T. D. Noe, Aug 10 2012 *)

from math import factorial
from sympy import binomial
def a(n):
    if n < 4:
        return 1
    else:
        return binomial(n, 3) * factorial(3*n-9) // factorial(2*n-4)
print([a(n) for n in range(3, 21)]) # Paul Muljadi, Mar 05 2024

a(n) = binomial(n,3)*(3*n-9)!/(2*n-4)!, n >= 4; a(3) = 1.

a(n) ~ 3^(3*n - 19/2) * n^(n-2) / (2^(2*n - 5/2) * exp(n)). - Vaclav Kotesovec, Mar 14 2024

More terms from Wolfdieter Lang

Name clarified by Andrey Zabolotskiy, Mar 15 2024

A079340 Absolute value of determinant of n X n matrix whose entries are the integers from 1 to n^2 spiraling outward, ending in a corner.

Original entry on oeis.org

1, 5, 72, 1380, 31920, 861840, 26611200, 925404480, 35805369600, 1526139014400, 71066912716800, 3590219977344000, 195589552648089600, 11430978821982720000, 713448513897799680000, 47363888351558338560000

Offset: 1

Kit Vongmahadlek (kit119(AT)yahoo.com), Jan 03 2003

If n == 0 or 1 (mod 4), the sign of the determinant will be independent of the orientation of the spiral. For n == 2 or 3 (mod 4), the sign will be reversed when the orientation is rotated by 1/4 or flipped on the horizontal or vertical axis. - Franklin T. Adams-Watters, Dec 31 2013

This distribution of the integers is sometimes known as Ulam's spiral, although that is sometimes reserved for when the primes are marked out in some way. - Franklin T. Adams-Watters, Dec 31 2013

			n=2, det=-5: {1 2 / 4 3 }.
n=3, det=72: {7 8 9 / 6 1 2 / 5 4 3 }.
n=4, det=-1380: { 7 8 9 10 / 6 1 2 11 /5 4 3 12 / 16 15 14 13 }.
n=5, det=31920: { 21 22 23 24 25 / 20 7 8 9 10 / 19 6 1 2 11 /18 5 4 3 12 / 17 16 15 14 13 }

Gaurav Bhatnagar, Christian Krattenthaler, Spiral determinants, arXiv:1704.02859 [math.CO], 2017.

Cf. A078475, A023999, A067276, A052182.

M[0, 0] = 1;
M[i_, j_] := If[i <= j,
  If[i + j >= 0, If[i != j, M[i + 1, j] + 1, M[i, j - 1] + 1],
   M[i, j + 1] + 1],
  If[i + j > 1, M[i, j - 1] + 1, M[i - 1, j] + 1]
  ]
M[n_] := If[EvenQ[n],
  Table[M[i, j], {j, n/2, -n/2 + 1, -1}, {i, -n/2 + 1, n/2}],
  Table[M[i, j], {j, (n - 1)/2, -(n - 1)/2, -1}, {i, -(n - 1)/2, (n - 1)/2}]]
a[n_]:=Det[M[n]] (* Christian Krattenthaler, Apr 19 2017 *)

A079340(n):=if n=1 then 1 else (2*n^2-3*n+3)*(2*n-2)!/(2*(n-1)!)$
makelist(A079340(n),n,1,30); /* Martin Ettl, Nov 05 2012 */

a(n) = (2*n^2-3*n+3) (2n-2)!/(2 (n-1)!) = A096376(n-1)*A000407(n-2), n>1. - Conjectured by Dean Hickerson, Jan 30 2003. Proved in the article by Bhatnagar and Krattenthaler.

D-finite with recurrence (2*n^2-7*n+8)*a(n) -2*(2*n-3)*(2*n^2-3*n+3)*a(n-1)=0. - R. J. Mathar, May 03 2019

Extended by Robert G. Wilson v, Jan 25 2003

A105725 Triangle read by rows: T(n,k)=(n+k)!/k! (0<=k<=n-1; n>=1).

Original entry on oeis.org

1, 2, 6, 6, 24, 60, 24, 120, 360, 840, 120, 720, 2520, 6720, 15120, 720, 5040, 20160, 60480, 151200, 332640, 5040, 40320, 181440, 604800, 1663200, 3991680, 8648640, 40320, 362880, 1814400, 6652800, 19958400, 51891840, 121080960, 259459200

Offset: 1

Amarnath Murthy, Apr 18 2005

T(n,n-1)=(2n-1)!/(n-1)! (A000407); T(n,0)=n! (A000142); Row sums yield A092956; Arithmetic means of the rows yield A001761.

Has many diagonals in common with A068424. - Zerinvary Lajos, Apr 14 2006

			1
2 6
6 24 60
24 120 360 840
120 720 2520 6720 15120
720 5040 20160 60480 151200 332640
5040 40320 181440 604800 1663200 3991680 8648640
40320 362880 1814400 6652800 19958400 51891840 121080960 259459200

Cf. A000407, A000142, A092956, A001761.

Maple
```
T:=proc(n,k) if k
				
```

T(n, k)=(n+k)!/k! (0<=k<=n-1; n>=1).

More terms from Emeric Deutsch, Apr 18 2005

A384024 a(n) = [x^n] Product_{k=0..n} (1 + (n+k)*x).

Original entry on oeis.org

1, 3, 26, 342, 5944, 127860, 3272688, 97053936, 3270729600, 123418922400, 5154170774400, 235977273544320, 11752173128586240, 632474276804697600, 36576553723886131200, 2261980049125982976000, 148956705206745595084800, 10406288081667512679321600, 768701832940487804295168000

Offset: 0

Vaclav Kotesovec, May 17 2025

nonn

Central terms of triangle A165675.

Cf. A000407, A201546, A383869.

Table[SeriesCoefficient[Product[1 + (n+k)*x, {k, 0, n}], {x, 0, n}], {n, 0, 20}]

a(n) = sum(k=0, n, (k+1)*n^k*abs(stirling(n+1, k+1, 1))); \\ Seiichi Manyama, May 18 2025

a(n) ~ n! * log(2) * 4^n * sqrt(n/Pi).
a(n) ~ log(2) * 2^(2*n + 1/2) * n^(n+1) / exp(n).
From Seiichi Manyama, May 18 2025: (Start)
a(n) = A165675(2*n,n).
a(n) = Sum_{k=0..n} (k+1) * n^k * |Stirling1(n+1,k+1)|.
a(n) = (n+1)! * Sum_{k=0..n} (-1)^k * binomial(-n,k)/(n+1-k).
a(n) = (2*n)!/n! * (1 + n * Sum_{k=1..n} 1/(n+k)). (End)

A086984 Number of arrangements of n labeled balls in n labeled columns where only 1 column may have more than 1 ball.

Original entry on oeis.org

1, 6, 60, 696, 9120, 134640, 2227680, 41005440, 833172480, 18546796800, 449223667200, 11766674304000, 331501679308800, 9997170543360000, 321355745238528000, 10969253822951424000, 396269940892041216000

Offset: 1

Jon Perry, Jul 27 2003

nonn

The difference between A000407 and A086984 is for example consider a(5). A000407 allows the 221 and 23 partitions, A086984 does not.

			a(2)=6;
.. .. -G -R R- G-
RG GR -R -G G- R-

Harvey P. Dale, Table of n, a(n) for n = 1..402

Cf. A000407, A086985.

Table[n!+Sum[Binomial[n-1,n-k],{k,2,n}]n n!,{n,20}] (* Harvey P. Dale, Nov 29 2019 *)

a(n)=n!+sum(i=2,n,binomial(n-1,n-i)*n*n!)

a(n) = n! + Sum_{i=2..n} binomial(n-1, n-i)*n*n!.

A091544 First column sequence of array A091746 ((6,2)-Stirling2).

Original entry on oeis.org

1, 30, 2700, 491400, 150368400, 69470200800, 45155630520000, 39285398552400000, 44078217175792800000, 61973973349164676800000, 106719182107261573449600000, 220908706962031457040672000000, 541226332056977069749646400000000, 1548989762347068373623487996800000000

Offset: 1

Wolfdieter Lang, Feb 13 2004

Also fifth column (m=4) sequence of triangle A091543.

Pawel Blasiak, Karol A. Penson, and Allan I. Solomon, The general boson normal ordering problem, Physics Letters A, Vol. 309, No. 3-4 (2003), pp. 198-205; arXiv preprint, arXiv:quant-ph/0402027, 2004.

Cf. A091535 (third column of A091543, first column of array A091534), A000407, A007696, A091746.

a[n_] := 2^(4*n-1) * Pochhammer[1/4, n] * Pochhammer[1/2, n]; Array[a, 20] (* Amiram Eldar, Aug 30 2025 *)

a(n) = 2^(n-1)*Product_{j=0..n-1}((2*j+1)*(4*j+1)), n>=1. From eq.12 of the Blasiak et al. reference with r=6, s=2, k=1.
a(n) = (2^(4*n-1))*risefac(1/4, n)*risefac(1/2, n), n>=1, with risefac(x, n) = Pochhammer(x, n).
a(n) = fac4(4*n-3)*fac4(4*n-2)/2, n>=1, with fac4(4*n-3) = A007696(n) and fac4(4*n-2)/2 = A000407(n+1) (quartic- or 4-factorials).
E.g.f.: (hypergeom([1/4, 1/2], [], 16*x)-1)/2.
a(n) = A091746(n, 2), n>=1.
a(n) ~ sqrt(Pi) * 2^(4*n) * n^(2*n-1/4) / (Gamma(1/4) * exp(2*n)). - Amiram Eldar, Aug 30 2025

A100622 Expansion of e.g.f. exp( (1+2x-sqrt(1-4x))/4).

Original entry on oeis.org

1, 1, 2, 10, 94, 1286, 22876, 499612, 12925340, 386356924, 13099953016, 496719289496, 20825694943912, 956599393819720, 47772070664027984, 2577034852683364816, 149335440671982405136, 9251650217381166689552, 610194993478502245703200, 42688019374465782644235424

Offset: 0

N. J. A. Sloane, Dec 04 2004

nonn

Number of topologically distinct solutions to the clone ordering problem for n clones.

			G.f. = 1 + x + 2*x^2 + 10*x^3 + 94*x^4 + 1286*x^5 + 22876*x^6 + 499612*x^7 + ...

Lee A. Newberg, Table of n, a(n) for n = 0..100
Lee Aaron Newberg, Finding, Evaluating and Counting DNA Physical Maps, Ph.D. Thesis, University of California, 1993, Berkeley, CA.
Lee A. Newberg, The Number of Clone Orderings, Discrete Applied Mathematics, Vol. 69 (1996) pp. 233-245.

E.g.f. (1+2*x-sqrt(1-4*x))/4 gives A000407.

Cf. A088218.

a := proc(n) option remember: if n = 0 then factorial(0) elif n = 1 then factorial(1) elif n = 2 then factorial(2) elif  n >= 3 then (4*n-5)*procname(n-1) - (4*n-7)*procname(n-2) + (n-2)*procname(n-3) fi; end:
seq(a(n), n = 0..250); # Muniru A Asiru, Jan 23 2018

CoefficientList[Series[Exp[(1+2*x-Sqrt[1-4*x])/4], {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jun 26 2013 *)

{a(n) = if( n<0, 0, n! * polcoeff( exp( (1 + 2*x - sqrt(1 - 4*x + x * O(x^n))) / 4), n))}; /* Michael Somos, Jan 03 2015 */

a(n) = n! for n = 0, 1, 2. a(n) = (4n-5) * a(n-1) - (4n-7) * a(n-2) + (n-2) * a(n-3) for n > 2. - Lee A. Newberg, Oct 18 2006
a(n) ~ n^(n-1)*exp(3/8-n)*2^(2*n-5/2). - Vaclav Kotesovec, Jun 26 2013
Given e.g.f. A(x), then A'(x) / A(x) = g.f. for A088218. - Michael Somos, Jan 03 2015
E.g.f.: exp( (1 + 2*x - sqrt(1 - 4*x)) / 4). - Michael Somos, Jan 03 2015
0 = a(n)*(+a(n+1) - 5*a(n+2) + 7*a(n+3) - a(n+4)) + a(n+1)*(-3*a(n+1) + 17*a(n+2) - 23*a(n+3) + 4*a(n+4)) + a(n+2)*(-8*a(n+2) + 12*a(n+3) - 4*a(n+4)) + a(n+3)*(+4*a(n+3)) for all n>-2. - Michael Somos, Jan 03 2015

cp's OEIS Frontend

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

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A295182 a(n) = n! * [x^n] exp(-n*x)/(1 - x)^n.

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A370915 A(n, k) = 4^n*Pochhammer(k/4, n). Square array read by ascending antidiagonals.

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A001762 Number of labeled n-vertex dissections of a ball.

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Julia

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A079340 Absolute value of determinant of n X n matrix whose entries are the integers from 1 to n^2 spiraling outward, ending in a corner.

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Maxima

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A105725 Triangle read by rows: T(n,k)=(n+k)!/k! (0<=k<=n-1; n>=1).

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Maple

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A384024 a(n) = [x^n] Product_{k=0..n} (1 + (n+k)*x).

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

A100622 Expansion of e.g.f. exp( (1+2x-sqrt(1-4x))/4).