A009445 -id:A009445 - cp's OEIS frontend

A214299 Triangle d_k(n) read by rows: number of n-th order Feynman diagrams with k interactions, 0<=k<=n.

1, 2, 4, 24, 16, 80, 720, 288, 480, 3552, 40320, 11520, 11520, 28416, 271104, 3628800, 806400, 576000, 852480, 2711040, 31342080, 479001600, 87091200, 48384000, 51148800, 97597440, 376104960, 5087692800, 87178291200

Offset: 0

R. J. Mathar, Jul 11 2012

			Triangle starts in row n=0 as:
1;
2,     4;
24,    16,    80;
720,   288,   480,   3552;
40320, 11520, 11520, 28416, 271104;

Alois P. Heinz, Rows n = 0..140, flattened
F. Battaglia, T. F. George, A Pascal type triangle for the number of topologically distinct many-electron Feynman diagrams, J. Math. Chem. 2 (1988) 241-247, triangle d_k(n).

Cf. A009445 (row sums), A214298 (main diagonal).

b:= proc(x, y, t) option remember; `if`(y>x or y<0, 0,
      `if`(x=0, 1, b(x-1, y-1, false)*`if`(t, (x+y)/y, 1) +
                   b(x-1, y+1, true)  ))
    end:
T:= (n, k) -> k!*2^k*b(2*k, 0, false)*binomial(n,k)*(2*n-2*k)!:
seq(seq(T(n,k), k=0..n), n=0..10);  # Alois P. Heinz, May 23 2015

b[x_, y_, t_] := b[x, y, t] = If[y>x || y<0, 0, If[x == 0, 1, b[x-1, y-1, False] * If[t, (x+y)/y, 1] + b[x-1, y+1, True]]]; T[n_, k_] := k!*2^k*b[2*k, 0, False] * Binomial[n, k]*(2*n - 2*k)!; Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* Jean-François Alcover, Jun 22 2015, after Alois P. Heinz *)

d_k(n) = A214298(k)*binomial(n,k)*(2n-2k)!.

A107252 a(n) = Product_{k=0..n-1} (n+k)!/(k+1)!.

Original entry on oeis.org

1, 1, 6, 1440, 36288000, 184354652160000, 309071606732292096000000, 254046582743105184577722777600000000, 142518177743863255019484504453155074867200000000000

Offset: 0

Henry Bottomley, May 14 2005

nonn

			a(3) = 1!*2!*3!*4!*5!/(1!*2!*3!*1!*2!) = 34560/24 = 1440.

Cf. A000178, A001700, A005130, A009963, A110131, A112332, A107254, A000142.

[1] cat [(&*[Factorial(n+k)/Factorial(k+1): k in [0..n-1]]): n in [1..10]]; // G. C. Greubel, May 21 2019

Table[Product[(n+k)!/(k+1)!,{k,0,n-1}],{n,0,10}] (* Alexander Adamchuk, Jul 10 2006 *)
a[n_] := (BarnesG[1 + 2 n] n! ((BarnesG[2 + n] Gamma[2 + n])/ BarnesG[3 + n])^(-1 + n))/BarnesG[2 + n]^2; Table[a[n], {n, 0, 10}] (* Peter Luschny, May 20 2019 *)

{a(n) = prod(k=0,n-1, (n+k)!/(k+1)!)}; \\ G. C. Greubel, May 21 2019

[product(factorial(n+k)/factorial(k+1) for k in (0..n-1)) for n in (0..10)] # G. C. Greubel, May 21 2019

a(n) = (n+1)!*(n+2)!*...*(2n-1)!/(1!*2!*...*(n-1)!).
a(n) = A000178(2n-1)/(A000178(n)*A000178(n-1)).
a(n) = A079478(n)/A001813(n).
a(n) = A079478(n-1)*A006963(n+1).
a(n) = A107251(n)/A000108(n).
a(n) = A107251(n-1)*A009445(n-1).
a(n) = A107254(n)/A000142(n).
a(n) = A009963(2n-1, n-1).
a(n) = A009963(2n-1, n).
a(n) = (G(1+2*n)*n!*((G(2+n)*Gamma(2+n))/G(3+n))^(n-1))/G(2+n)^2, where G(x) is the Barnes G function. - Peter Luschny, May 20 2019
a(n) ~ A * 2^(2*n^2 - 7/12) * n^(n^2 - n - 5/12) / (sqrt(Pi) * exp(3*n^2/2 - n + 1/12)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, May 21 2019

A138896 Ratio of (2n-1)! to number of zeros in Sylvester matrix of polynomial of n degree with all nonzero coefficients.

Original entry on oeis.org

3, 15, 280, 11340, 798336, 86486400, 13343616000, 2778808032000, 750895681536000, 255454710858547200, 106826515449937920000, 53858368206010368000000, 32215590089995124736000000, 22555515290152300904448000000, 18272974787062050706056806400000, 16959604724241965811558973440000000

Offset: 2

Artur Jasinski, Apr 02 2008

nonn

(2n-1)! = A009445(n-1) is the number of monomials in determinant of symbolic square matrix of size 2n-1 X 2n-1 without zeros.

Denominators in the series expansion of (1/2)*(Pi/(2*x))^(1/2)* (x*BesselI(1/2, x) - BesselI(3/2, x)). - Abdallah Daddi-Moussa-Ider, Jul 25 2024

Cf. A001105, A009445, A007878, A138897, A138898.

Table[(2 n - 1)!/(2 (n - 1)^2), {n, 2, 20}]

a(n) = (2*n - 1)!/(2*(n - 1)^2).
Sum_{n=2..oo} 1/a(n) = (e^2 - 3)/(4*e) = 0.40366087623617955676434290... . - Stefano Spezia, Jul 25 2024, simplified by Vaclav Kotesovec, Aug 19 2025
Sum_{n>=2} (-1)^n/a(n) = cos(1)/2. - Amiram Eldar, Aug 19 2025

a(15)-a(17) from Stefano Spezia, Jul 25 2024

A138897 Ratio of (2n-1)! to number of zeros in upper part of Sylvester matrix of polynomial of degree n with all nonzero coefficients.

Original entry on oeis.org

3, 20, 420, 18144, 1330560, 148262400, 23351328000, 4940103168000, 1351612226764800, 464463110651904000, 195848611658219520000, 99430833611096064000000, 59828953024276660224000000, 42103628541617628354969600000, 34261827725741345073856512000000, 31923961833867229762934538240000000

Offset: 2

Artur Jasinski, Apr 02 2008

nonn

From Anthony Hernandez, Oct 24 2017: (Start)
If (n,n-1) is the two-part partition of any odd integer greater than 1 then a(n-1) is the number of permutations of shape (n,n-1). For example, the two-part partition of 11 with shape (n,n-1) is (6,5). Pictorially we can draw this as a standard Young diagram with cells populated by hook lengths:
          (6,5) = 7 6 5 4 3 1
                  5 4 3 2 1
  and there are a(6-1) = a(5) = 1330560 permutations with shape (6,5). (End)

Cf. A002378, A009445, A007878, A138896, A138898, A000108.

A138897:=n->(2*n - 1)!/(n*(n - 1)): seq(A138897(n), n=2..20); # Wesley Ivan Hurt, Nov 25 2017

Table[(2 n - 1)!/(n (n - 1)), {n, 2, 20}]

a(n) = (2*n - 1)!/(n*(n - 1)); \\ Michel Marcus, Oct 28 2017

a(n) = (2n - 1)!/(n*(n - 1)).
Sum_{n>=2} 1/a(n) = (1 + e^2)/(8*e) = 0.38577015870381094461947640518926542... . - Stefano Spezia, Jul 27 2024
Sum_{n>=2} (-1)^n/a(n) = (2*sin(1) - cos(1))/4. - Amiram Eldar, Aug 19 2025

More terms from Michel Marcus, Oct 28 2017

A138898 Ratio of (2*n-1)! to number of zeros in lower part of Sylvester matrix for polynomial of degree n with all nonzero coefficients.

Original entry on oeis.org

60, 840, 30240, 1995840, 207567360, 31135104000, 6351561216000, 1689515283456000, 567677135241216000, 235018333989863424000, 117509166994931712000000, 69800445194989436928000000, 48581109855712648101888000000, 39156374543704394370121728000000, 36180490078382860397992476672000000, 37989514582302003417892100505600000000, 44979585265445572046784246998630400000000, 59642930061980828534035911520183910400000000

Offset: 3

Artur Jasinski, Apr 02 2008

Cf. A002378, A009445, A007878, A138896.

Table[(2 n - 1)!/((n - 1)(n - 2)), {n, 3, 20}] (* R. J. Mathar, Apr 30 2008 *)

a(n) = (2*n-1)!/((n-1)*(n-2)). - R. J. Mathar, Apr 30 2008
Sum_{n=3..oo} 1/a(n) = (3*cosh(1) + 10*sinh(1) - 6*e)/4 = 0.0178907175323686230239526350278045532... . - Stefano Spezia, Jul 27 2024
Equivalently, Sum_{n=3..oo} 1/a(n) = (e^2 - 7)/(8*e). - Vaclav Kotesovec, Aug 19 2025
Sum_{n>=3} (-1)^(n+1)/a(n) = (2*sin(1) - 3*cos(1))/4. - Amiram Eldar, Aug 19 2025

Edited and corrected by R. J. Mathar, Apr 30 2008

A256881 a(n) = n!/ceiling(n/2).

Original entry on oeis.org

1, 2, 3, 12, 40, 240, 1260, 10080, 72576, 725760, 6652800, 79833600, 889574400, 12454041600, 163459296000, 2615348736000, 39520825344000, 711374856192000, 12164510040883200, 243290200817664000, 4644631106519040000, 102181884343418880000, 2154334728240414720000

Offset: 1

M. F. Hasler, Apr 22 2015

nonn

Original name was: n!/round(n/2). - Robert Israel, Sep 03 2018

Robert Israel, Table of n, a(n) for n = 1..450
Pierre-Alain Sallard, Sum of repeated integrals of sinh.

Cf. A009445, A052612, A052616, A052849, A081457, A208529, A098558, A107991, A110468, A229244, A256031.

[Factorial(n)/Round(n/2): n in [1..30]]; // Vincenzo Librandi, Apr 23 2015

Maple
```
A256881 := n!/round(n/2);
```

Function[x, 1/x] /@
CoefficientList[Series[(Sinh[x] + x*Exp[x])/2, {x, 0, 20}], x] (* Pierre-Alain Sallard, Dec 15 2018 *)

PARI
```
A256881(n)=n!/round(n/2)
```

a(2n) = 2*A009445(n) = A052612(2n-1) = A052616(2n-1) = A052849(2n-1) = A098558(2n-1) = A081457(3n-1) = A208529(2n+1) = A256031(2n-1).
a(2n+1) = A110468(n) = A107991(2n+2) = A229244(2n+1), n>=0.
From Robert Israel, Sep 03 2018: (Start)
E.g.f.: -(1+1/x)*log(1-x^2).
n*(n+1)*(n+2)*a(n)+(n+2)*a(n+1)-(n+3)*a(n+2)=0. (End)
a(n) = 2/([x^n](sinh(x) + x*exp(x))). - Pierre-Alain Sallard, Dec 15 2018
Sum_{n>=1} 1/a(n) = (3*e-1/e)/4 = (e + sinh(1))/2. - Amiram Eldar, Feb 02 2023

Definition clarified by Robert Israel, Sep 03 2018

A334378 Decimal expansion of Sum_{k>=0} 1/((2*k+1)!)^2.

Original entry on oeis.org

1, 0, 2, 7, 8, 4, 7, 2, 6, 1, 5, 9, 7, 4, 1, 5, 7, 9, 9, 6, 9, 2, 6, 8, 8, 4, 9, 3, 0, 8, 0, 7, 9, 2, 3, 6, 3, 7, 3, 0, 3, 4, 3, 3, 1, 0, 2, 8, 3, 4, 2, 5, 7, 2, 5, 4, 7, 1, 2, 4, 5, 0, 2, 2, 8, 2, 6, 7, 2, 5, 6, 9, 2, 7, 3, 2, 3, 3, 2, 8, 1, 8, 8, 5, 7, 3, 5, 2, 7, 8, 8, 3, 5, 1, 5, 2, 8, 2, 6, 6, 4, 6, 7, 6, 7, 9, 2, 3, 7, 8

Offset: 1

Ilya Gutkovskiy, Apr 25 2020

			1/1!^2 + 1/3!^2 + 1/5!^2 + 1/7!^2 + ... = 1.027847261597415799692...
Continued fraction: 1 + 1/(36 - 36/(401 - 400/(1765 - ... - P(n-1)/((P(n) + 1) - ... )))), where P(n) = (2*n*(2*n + 1))^2 for n >= 1. - _Peter Bala_, Feb 22 2024

Index entries for sequences related to Bessel functions or polynomials

Cf. A009445, A049469, A070910, A073742, A091681, A197036, A334379.

RealDigits[(BesselI[0, 2] - BesselJ[0, 2])/2, 10, 110] [[1]]

suminf(k=0, 1/((2*k+1)!)^2) \\ Michel Marcus, Apr 26 2020

(besseli(0,2) - besselj(0,2))/2 \\ Michel Marcus, Apr 26 2020

Equals (BesselI(0,2) - BesselJ(0,2))/2.

A204420 Triangle T(n,k) giving number of degree-2n permutations which decompose into exactly k cycles of even length, k=0..n.

Original entry on oeis.org

1, 0, 1, 0, 6, 3, 0, 120, 90, 15, 0, 5040, 4620, 1260, 105, 0, 362880, 378000, 132300, 18900, 945, 0, 39916800, 45571680, 18711000, 3534300, 311850, 10395, 0, 6227020800, 7628100480, 3511347840, 794593800, 94594500, 5675670, 135135

Offset: 0

José H. Nieto S., Jan 15 2012

The row polynomials t(n,x):= sum(T(n,k)*x^k, k=0..n) satisfy the recurrence relation t(n,x) = (2n-1)*(x+2n-2)*t(n-1,x), with t(0,x)=1, hence t(n,x)=(2n-1)!!*x(x+2)(x+4)...(x+2n-2).

			1;
0,        1,
0,        6,        3;
0,      120,       90,       15;
0,     5040,     4620,     1260,     105;
0,   362880,   378000,   132300,   18900,    945;
0, 39916800, 45571680, 18711000, 3534300, 311850, 10395;

Alois P. Heinz, Rows n = 0..85, flattened
Steven Finch, Rounds, Color, Parity, Squares, arXiv:2111.14487 [math.CO], 2021.
Angelo Lucia and Amanda Young, A Nonvanishing Spectral Gap for AKLT Models on Generalized Decorated Graphs, arXiv:2212.11872 [math-ph], 2022.

Cf. A049218, A060523, A060524, A132393, A048994.

Row sums give: A001818. - Alois P. Heinz, Jul 21 2013

T_row:= proc(n) local k; seq(doublefactorial(2*n-1)*2^(n-k)* coeff(expand(pochhammer(x, n)), x, k), k=0..n) end: seq(T_row(n), n=0..10);

nn=12;Prepend[Map[Prepend[Select[#,#>0&],0]&,Table[(Range[0, nn]!CoefficientList[ Series[(1-x^2)^(-y/2),{x,0,nn}],{x,y}])[[n]],{n,3,nn,2}]],{1}]//Grid (* Geoffrey Critzer, Jul 21 2013 *)

T(n,k) = (2*n)!/(2^n*n!)*(-2)^(n-k)*stirling(n,k,1); \\ Andrew Howroyd, Feb 12 2018

T(n,k) = (2n-1)!!*2^(n-k)*A132393(n,k).
T(n,k) = (2n-1)T(n-1,k-1) + (2n-1)(2n-2)*T(n-1,k); T(0,0)=1, T(n,0)=0 for n>0,
T(n,n) = (2n-1)!! = A001147(n).
T(n,1) = (2n-1)! = A009445(n-1).

A226731 a(n) = (2n - 1)!/(2n).

Original entry on oeis.org

20, 630, 36288, 3326400, 444787200, 81729648000, 19760412672000, 6082255020441600, 2322315553259520000, 1077167364120207360000, 596585001666576384000000, 388888194657798291456000000

Offset: 3

Wesley Ivan Hurt, Jun 15 2013

For n < 3, the formula does not produce an integer.

The ratio of the product of the partition parts of 2n into exactly two parts to the sum of the partition parts of 2n into exactly two parts. For example, a(3) = 20, and 2*3 = 6 has 3 partitions into exactly two parts: (5,1), (4,2), (3,3). Forming the ratio of product to sum (of parts), we have (5*1*4*2*3*3)/(5+1+4+2+3+3) = 360/18 = 20. - Wesley Ivan Hurt, Jun 24 2013

			a(3) = (2*3 - 1)!/(2*3) = 5!/6 = 120/6 = 20.

Seiichi Manyama, Table of n, a(n) for n = 3..225
Index entries for sequences related to partitions.

Cf. A001105, A002674, A005843, A009445, A093353, A143623, A211374.

seq((2*k-1)!/(2*k),k=1..20); # Wesley Ivan Hurt, Jun 24 2013

Table[(2n-1)!/(2n),{n,3,20}] (* Harvey P. Dale, Jun 19 2013 *)

a(n) = (2*n-1)!/(2*n); \\ Michel Marcus, Nov 01 2016

a(n) = A009445(n-1)/A005843(n) = A002674(n)/A001105(n). - Wesley Ivan Hurt, Jun 24 2013
a(n) ~ sqrt(Pi)*2^(2*n-1)*n^(2*n-3/2)/exp(2*n). - Ilya Gutkovskiy, Nov 01 2016
From Amiram Eldar, Mar 10 2022: (Start)
Sum_{n>=3} 1/a(n) = e - 8/3.
Sum_{n>=3} (-1)^(n+1)/a(n) = cos(1) + sin(1) - 4/3. (End)

A245602 Triangle read by rows: the negative terms of A163626.

Original entry on oeis.org

-1, -3, -7, -6, -15, -60, -31, -390, -120, -63, -2100, -2520, -127, -10206, -31920, -5040, -255, -46620, -317520, -181440, -511, -204630, -2739240, -3780000, -362880, -1023, -874500, -21538440, -59875200, -19958400, -2047

Offset: 0

Paul Curtz, Dec 17 2014

These numbers a(n) are the companion of A249163(n).
Consider the Worpitzky fractions A163626(n)/A002260(n) yielding the second Bernoulli numbers A164555(n)/A027642(n):
1,
1, -1/2,
1, -3/2,  +2/3,
1, -7/2, +12/3, -6/4,
etc.
From the second row on, the sum of the numerators is 0.
The absolute values of every row of the numerators triangle A163626 are 1, 2, 6, 26, ... = A000629(n).
a(n) triangle is shifted. It starts from second row and second column of triangle above.
-1,
-3,
-7,       -6,
-15,     -60,
-31,    -390,    -120,
-63,   -2100,   -2520,
-127, -10206,  -31920,   -5040,
-255, -46620, -317520, -181440,
etc.
Sum of successive rows: -1, -3, -13, -75, ... = -A000670(n+1).
Successive columns: A000225, A028244, from the Stirling numbers of second kind S(n,2), S(n,4), S(n,6), S(n,8), S(n,10), ... . See A000770, A032180, A049434, A228910, A049435, A228912, A008277.
Thanks to Jean-François Alcover.

Cf. A000225, A000629, A000670, A000770, A002260, A008277, A009445, A027642, A028244, A032180, A049434, A049435, A163626, A164555, A228910, A228912, A249163.

Select[ Table[ (-1)^k*k!*StirlingS2[n+1, k+1], {n, 0, 12}, {k, 0, n}] // Flatten, Negative] (* Jean-François Alcover, Dec 26 2014 *)

cp's OEIS Frontend

A214299 Triangle d_k(n) read by rows: number of n-th order Feynman diagrams with k interactions, 0<=k<=n.

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A107252 a(n) = Product_{k=0..n-1} (n+k)!/(k+1)!.

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A138896 Ratio of (2n-1)! to number of zeros in Sylvester matrix of polynomial of n degree with all nonzero coefficients.

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A138897 Ratio of (2n-1)! to number of zeros in upper part of Sylvester matrix of polynomial of degree n with all nonzero coefficients.

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A138898 Ratio of (2*n-1)! to number of zeros in lower part of Sylvester matrix for polynomial of degree n with all nonzero coefficients.

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A256881 a(n) = n!/ceiling(n/2).

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A334378 Decimal expansion of Sum_{k>=0} 1/((2*k+1)!)^2.

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