A090438 -id:A090438 - cp's OEIS frontend

A090439 Alternating row sums of array A090438 ((4,2)-Stirling2).

1, 5, 37, -887, -168919, -21607859, -2799384755, -337767590383, -11912361112367, 21032925955607701, 16703816669710968821, 10654267957172226744985, 6614425802684094455696377, 4120103872599589439389105373

Offset: 1

Wolfdieter Lang, Dec 23 2003

# assuming offset 0:
p := (n,x) -> (2*n+2)!*hypergeom([-2*n],[3],x)/2;
seq(simplify(p(n,1)), n=0..11); # Peter Luschny, Apr 08 2015

a[n_, k_] := (-1)^k/k! Sum[(-1)^p Binomial[k, p] Product[FactorialPower[p + 2(j-1), 2], {j, 1, n}], {p, 2, k}];
a[n_] := Sum[(-1)^k a[n,k], {k, 2, 2n}];
Array[a,14] (* Jean-François Alcover, Jun 05 2019 *)

a(n) = Sum_{k=2..2*n} ((-1)^k)*A090438(n, k), n>=1, a(0):= 1.

a(n) = (2*n+2)!*hypergeom([-2*n],[3],1)/2, assuming offset 0. - Peter Luschny, Apr 08 2015

A091032 Second column (k=3) of array A090438 ((4,2)-Stirling2) divided by 8.

Original entry on oeis.org

1, 60, 5040, 604800, 99792000, 21794572800, 6102480384000, 2134124568576000, 912338253066240000, 468333636574003200000, 284372184127734743040000, 201645730563302817792000000, 165147853331345007771648000000

Offset: 2

Wolfdieter Lang, Jan 23 2004

John Campbell, Applications of Gosper’s nonlocal derangement identity, Maple Transactions, 5, 1, Article 16724, Feb. 2025.

Cf. A002674 (first column of A090438), A091033 (third column), A090438.

Cf. A001620, A049470, A073742, A073743, A099281, A099282, A099284.

a[n_] := (n - 1)*(2*n)!/4!; Array[a, 13, 2] (* Amiram Eldar, Nov 03 2022 *)

a(n) = (n-1)*(2*n)!/4!; \\ Amiram Eldar, Nov 03 2022

a(n) = A090438(n, 3)/8 = (n-1)*(2*n)!/4!
E.g.f.: (-3*hypergeom([1/2, 1], [], 4*x) + hypergeom([1, 3/2], [], 4*x) + 2)/(8*3!) (cf. A090438).
From Amiram Eldar, Nov 03 2022: (Start)
Sum_{n>=2} 1/a(n) = 60 - 24*Gamma - 24*cosh(1) + 24*CoshIntegral(1) - 24*sinh(1).
Sum_{n>=2} (-1)^n/a(n) = -12 + 24*gamma - 24*cos(1) - 24*CosIntegral(1) + 24*SinIntegral(1). (End)
a(n+1) = Sum_{j = 1..n} (-1)^(n+j) * j^(2*n+2) * binomial(2*n, n-j) (Campbell, Eq. 17). - Peter Bala, Mar 30 2025

A091033 Third column (k=4) of array A090438 ((4,2)-Stirling2).

Original entry on oeis.org

1, 180, 25200, 4233600, 898128000, 239740300800, 79332244992000, 32011868528640000, 15509750302126080000, 8898339094906060800000, 5971815866682429603840000, 4637851802955964809216000000

Offset: 2

Wolfdieter Lang, Jan 23 2004

Cf. A091032 (second column of A090438 divided by 8), A091034 (fourth column divided by 24), A000384, A090438.

Cf. A001620, A049469, A049470, A073742, A099281, A099282, A099283, A099284.

a[n_] := (n-1)*(2*n-3)*(2*n)!/4!; Array[a, 12, 2] (* Amiram Eldar, Nov 03 2022 *)

a(n) = (n-1)*(2*n-3)*(2*n)!/4!; \\ Amiram Eldar, Nov 03 2022

a(n) = A090438(n, 4), n>=2.
a(n) = (n-1)*(2*n-3)*(2*n)!/4! = binomial(2*(n-1), 2)*(2*n)!/4! = A000384(n-1)*(2*n)!/4!, n>=2.
E.g.f.: (6*hypergeom([1/2, 1], [], 4*x) - 4*hypergeom([1, 3/2], [], 4*x) + hypergeom([3/2, 2], [], 4*x) -3)/4! (cf. A090438).
From Amiram Eldar, Nov 03 2022: (Start)
Sum_{n>=2} 1/a(n) = -20 + 24*Gamma - 16*CoshIntegral(1) + 16*sinh(1) + 8*SinhIntegral(1).
Sum_{n>=2} (-1)^n/a(n) = 4 - 24*gamma + 16*cos(1) + 24*CosIntegral(1) - 16*sin(1) + 8*SinIntegral(1). (End)

A091034 Fourth column (k=5) of array A090438 ((4,2)-Stirling2) divided by 24.

Original entry on oeis.org

1, 280, 70560, 19958400, 6659452800, 2644408166400, 1244905998336000, 689322235650048000, 444916954745303040000, 331767548149023866880000, 283424276847308960563200000, 275246422218908346286080000000

Offset: 3

Wolfdieter Lang, Jan 23 2004

Cf. A091033 (third column of A090438), A091035 (fifth column), A090438.

Cf. A001620, A049469, A049470, A073742, A073743, A099281, A099282, A099283, A099284.

a[n_] := (n - 1)*(n - 2)*(2*n - 3)*(2*n)!/(5!*(3!)^2); Array[a, 12, 3] (* Amiram Eldar, Nov 03 2022 *)

a(n) = (n-1)*(n-2)*(2*n-3)*(2*n)!/(5!*(3!)^2); \\ Amiram Eldar, Nov 03 2022

a(n) = A090438(n, 5)/24, n>=3.
a(n) = (n-1)*(n-2)*(2*n-3)*(2*n)!/(5!*(3!)^2), n>=3.
E.g.f.: (Sum_{p=2..5} (((-1)^(p+1))*binomial(5, p)*hypergeom([(p-1)/2, p/2], [], 4*x)) + 4)/(5!*4!) (cf. A090438).
From Amiram Eldar, Nov 03 2022: (Start)
Sum_{n>=3} 1/a(n) = 2010 - 4680*Gamma + 1800*cosh(1) + 4680*CoshIntegral(1) - 2520*sinh(1) - 2880*SinhIntegral(1).
Sum_{n>=3} (-1)^(n+1)/a(n) = -2010 - 3960*gamma + 3240*cos(1) + 3960*CosIntegral(1) - 1800*sin(1) + 2880*SinIntegral(1). (End)

A091035 Fifth column (k=6) of array A090438 ((4,2)-Stirling2).

Original entry on oeis.org

1, 840, 352800, 139708800, 59935075200, 29088489830400, 16183777978368000, 10339833534750720000, 7563588230670151680000, 6303583414831453470720000, 5951909813793488171827200000, 6330667711034891964579840000000

Offset: 3

Wolfdieter Lang, Jan 23 2004

Cf. A091034 (fourth column of A090438 divided by 24), A091036 (sixth column divided by 48), A053134, A090438.

Cf. A001620, A049469, A049470, A073742, A073743, A099281, A099282, A099283, A099284.

Table[Binomial[2n-2,4] (2n)!/6!,{n,3,20}] (* Harvey P. Dale, Jun 07 2021 *)

a(n) = binomial(2*n-2, 4)*(2*n)!/6!; \\ Amiram Eldar, Nov 03 2022

a(n) = A090438(n, 6), n>=3.
a(n) = binomial(2*n-2, 4)*(2*n)!/6! = A053134(n-3)*(2*n)!/6!, n>=3.
E.g.f.: (Sum_{p=2..6} (((-1)^p)*binomial(6, p)*hypergeom([(p-1)/2, p/2], [], 4*x)) - 5)/6! (cf. A090438).
From Amiram Eldar, Nov 03 2022: (Start)
Sum_{n>=3} 1/a(n) = -594 + 1800*Gamma - 1008*cosh(1) - 1800*CoshIntegral(1) + 912*sinh(1) + 1464*SinhIntegral(1).
Sum_{n>=3} (-1)^(n+1)/a(n) = 1554 + 1080*gamma - 1248*cos(1) - 1080*CosIntegral(1) + 240*sin(1) - 1416*SinIntegral(1). (End)

A091036 Sixth column (k=7) of array A090438 ((4,2)-Stirling2) divided by 48=4!*2.

Original entry on oeis.org

1, 840, 498960, 285405120, 173145772800, 115598414131200, 86165279456256000, 72034173625430016000, 67538393730337001472000, 70856069211827240140800000, 82901600977837870964736000000

Offset: 4

Wolfdieter Lang, Jan 23 2004

Cf. A091035 (fifth column of A090438).

A091036 := proc(n)
    binomial(2*n-2,5)*(2*n)!/7!/4!/2 ;
end proc:
seq(A091036(n),n=4..40) ; # R. J. Mathar, Jul 27 2022

a(n)=A090438(n, 7)/48, n>=4.
a(n)=binomial(2*n-2, 5)*(2*n)!/(7!*4!*2)= A053132(n+1)*(2*n)!/(7!*4!), n>=4.
E.g.f.:(sum(((-1)^(p+1))*binomial(7, p)*hypergeom([(p-1)/2, p/2], [], 4*x), p=2..7) + 6)/(7!*48) (cf. A090438).
D-finite with recurrence (2*n-7)*(n-4)*a(n) -2*n*(n-1)*(2*n-1)*(2*n-3)*a(n-1)=0. - R. J. Mathar, Jul 27 2022

A002674 a(n) = (2n)!/2.

Original entry on oeis.org

1, 12, 360, 20160, 1814400, 239500800, 43589145600, 10461394944000, 3201186852864000, 1216451004088320000, 562000363888803840000, 310224200866619719680000, 201645730563302817792000000, 152444172305856930250752000000, 132626429906095529318154240000000

Offset: 1

N. J. A. Sloane, Simon Plouffe

Right side of the binomial sum n-> sum( (-1)^i * (n-i)^(2*n) * binomial(2*n, i), i=0..n). - Yong Kong (ykong(AT)curagen.com), Dec 28 2000
a(n) is the number of ways to display n distinct flags on n distinct poles and then linearly order all (including any empty) poles. - Geoffrey Critzer, Dec 16 2009
Product of the partition parts of 2n into exactly two parts. - Wesley Ivan Hurt, Jun 03 2013
Let f(x) be a polynomial in x. The expansion (2*sinh(x/2))^2 = x^2 + (1/12)*x^4 + (1/360)*x^6 + ... leads to the second central difference formula f(x+1) - 2*f(x) + f(x-1) = (2*sinh(D/2))^2(f(x)) = D^2(f(x)) + (1/12)*D^4(f(x)) + (1/360)* D^6(f(x)) + ..., where D denotes the differential operator d/dx. - Peter Bala, Oct 03 2019

			a(3) = 360, since 2(3) = 6 has exactly 3 partitions into two parts: (5,1), (4,2), (3,3).  Multiplying all the parts in the partitions, we get 5! * 3 = 360. - _Wesley Ivan Hurt_, Jun 03 2013

A. P. Prudnikov, Yu. A. Brychkov and O.I. Marichev, "Integrals and Series", Volume 1: "Elementary Functions", Chapter 4: "Finite Sums", New York, Gordon and Breach Science Publishers, 1986-1992, Eq. (4.2.2.33)
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).

Vincenzo Librandi, Table of n, a(n) for n = 1..100
Ronald P. Nordgren, Compound Lucas Magic Squares, arXiv:2103.04774 [math.GM], 2021. See Table 2 p. 12.
H. E. Salzer, Tables of coefficients for obtaining central differences from the derivatives, Journal of Mathematics and Physics (this journal is also called Studies in Applied Mathematics), 42 (1963), 162-165, plus several inserted tables.
H. E. Salzer, Annotated scanned copy of left side of Table II.
Eric Weisstein's World of Mathematics, Central Difference.

a(n) = A090438(n, 2), n >= 1 (first column of (4, 2)-Stirling2 array).

Cf. A000142, A010050, A090438, A002544. Cf. A002671, A002672, A002673, A002675, A002676, A002677.

[n*Factorial(2*n-1): n in [1..15]]; // Vincenzo Librandi, Aug 23 2013

seq((2*k)!/2, k=1..20); # Wesley Ivan Hurt, Aug 22 2013

Table[n! Pochhammer[n, n], {n, 0, 10}] (* Geoffrey Critzer, Dec 16 2009 *)
Table[(2 n)! / 2, {n, 1, 15}] (* Vincenzo Librandi, Aug 23 2013 *)

a(n) = (2*n)!/2; \\ Indranil Ghosh, Apr 18 2017

4*sinh(x/2)^2 = Sum_{k>=1} x^(2k)/a(k). - Benoit Cloitre, Dec 08 2002
E.g.f.: (hypergeom([1/2, 1], [], 4*x)-1)/2 (cf. A090438).
a(n) = n*(2n-1)!. - Geoffrey Critzer, Dec 16 2009
a(n) = A010050(n)/2. - Wesley Ivan Hurt, Aug 22 2013
a(n) = Product_{k=0..n-1} (n^2 - k^2). - Stanislav Sykora, Jul 14 2014
Series reversion ( Sum_{n >= 1} x^n/a(n) ) = Sum_{n >= 1} (-1)^n*x^n/b(n-1), where b(n) = A002544(n). - Peter Bala, Apr 18 2017
From Amiram Eldar, Jul 09 2020: (Start)
Sum_{n>=1} 1/a(n) = 2*(cosh(1) - 1).
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*(1 - cos(1)). (End)

A091534 Generalized Stirling2 array (5,2).

Original entry on oeis.org

1, 20, 10, 1, 1120, 1040, 290, 30, 1, 123200, 161920, 71320, 14040, 1340, 60, 1, 22422400, 37452800, 22097600, 6263040, 958720, 82800, 4000, 100, 1, 6098892800, 12222918400, 8928102400, 3257116800, 675281600, 84782880, 6625920, 322000

Offset: 1

Wolfdieter Lang, Jan 23 2004

The row length sequences for this array is [1,3,5,7,9,11,...]=A005408(n-1), n>=1.

P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, Phys. Lett. A 309 (2003) 198-205.
W. Lang, First 6 rows.
M. Schork, On the combinatorics of normal ordering bosonic operators and deforming it, J. Phys. A 36 (2003) 4651-4665.

Cf. A078740 (3, 2)-Stirling2, A090438 (4, 2)-Stirling2.

Cf. A072019 (row sums), A091537 (alternating row sums).

a[n_, k_] := (-1)^k/k!*Sum[(-1)^p*Binomial[k, p]*Product[FactorialPower[p + 3*(j - 1), 2], {j, 1, n}], {p, 2, k}]; Table[a[n, k], {n, 1, 8}, {k, 2, 2 n}] // Flatten (* Jean-François Alcover, Sep 01 2016 *)

a(n, k)=(((-1)^k)/k!)*sum(((-1)^p)*binomial(k, p)*product(fallfac(p+3*(j-1), 2), j=1..n), p=2..k), n>=1, 2<=k<=2*n, else 0. From eq. (12) of the Blasiak et al. reference with r=5, s=2.

Recursion: a(n, k)=sum(binomial(2, p)*fallfac(3*(n-1)+k-p, 2-p)*a(n-1, k-p), p=0..2), n>=2, 2<=k<=2*n, a(1, 2)=1, else 0. Rewritten from eq.(19) of the Schork reference with r=5, s=2. fallfac(n, m) := A008279(n, m) (falling factorials triangle).

A091746 Generalized Stirling2 array (6,2).

Original entry on oeis.org

1, 30, 12, 1, 2700, 1920, 426, 36, 1, 491400, 478800, 162540, 25344, 1956, 72, 1, 150368400, 181440000, 80451000, 17624880, 2130660, 147840, 5820, 120, 1, 69470200800, 98424849600, 52905560400, 14618016000, 2346624000, 232202880

Offset: 1

Wolfdieter Lang, Feb 27 2004

The sequence of row lengths for this array is [1,3,5,7,9,11,...]=A005408(n-1), n>=1.

P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, Phys. Lett. A 309 (2003) 198-205.
M. Schork, On the combinatorics of normal ordering bosonic operators and deforming it, J. Phys. A 36 (2003) 4651-4665.

W. Lang, First 6 rows.

Cf. A078740 (3, 2)-Stirling2, A090438 (4, 2)-Stirling2, A091534 (5, 2)-Stirling2.
Cf. A091544 (first column), A091550 (second column divided by 12).
Cf. A091748 (row sums), A091750 (alternating row sums).

a[n_, k_] := (-1)^k/k! Sum[(-1)^p Binomial[k, p] Product[FactorialPower[p + 4*(j-1), 2], {j, 1, n}], {p, 2, k}]; Table[a[n, k], {n, 1, 8}, {k, 2, 2n} ] // Flatten (* Jean-François Alcover, Sep 01 2016 *)

a(n, k)=(((-1)^k)/k!)*sum(((-1)^p)*binomial(k, p)*product(fallfac(p+4*(j-1), 2), j=1..n), p=2..k), n>=1, 2<=k<=2*n, else 0. From eq. (12) of the Blasiak et al. reference with r=6, s=2.

Recursion: a(n, k)=sum(binomial(2, p)*fallfac(4*(n-1)+k-p, 2-p)*a(n-1, k-p), p=0..2), n>=2, 2<=k<=2*n, a(1, 2)=1, else 0. Rewritten from eq.(19) of the Schork reference with r=6, s=2. fallfac(n, m) := A008279(n, m) (falling factorials triangle).

A091747 Generalized Stirling2 array (7,2).

Original entry on oeis.org

1, 42, 14, 1, 5544, 3192, 588, 42, 1, 1507968, 1165248, 321552, 41496, 2688, 84, 1, 696681216, 655966080, 232606080, 41497344, 4143552, 240240, 7980, 140, 1, 489070213632, 533531142144, 226306918656, 50249808000, 6575950080

Offset: 1

Wolfdieter Lang, Feb 27 2004

The sequence of row lengths for this array is [1,3,5,7,9,11,...]=A005408(n-1), n>=1.

P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, Phys. Lett. A 309 (2003) 198-205.
M. Schork, On the combinatorics of normal ordering bosonic operators and deforming it, J. Phys. A 36 (2003) 4651-4665.

W. Lang, First 6 rows.

Cf. A078740 (3, 2)-Stirling2, A090438 (4, 2)-Stirling2, A091534 (5, 2)-Stirling2, A091746 (6, 2)-Stirling2.
Cf. A091545 (first column).
Cf. A091749 (row sums), A091751 (alternating row sums).

a(n, k)=(((-1)^k)/k!)*sum(((-1)^p)*binomial(k, p)*product(fallfac(p+5*(j-1), 2), j=1..n), p=2..k), n>=1, 2<=k<=2*n, else 0. From eq. (12) of the Blasiak et al. reference with r=7, s=2.

Recursion: a(n, k)=sum(binomial(2, p)*fallfac(5*(n-1)+k-p, 2-p)*a(n-1, k-p), p=0..2), n>=2, 2<=k<=2*n, a(1, 2)=1, else 0. Rewritten from eq.(19) of the Schork reference with r=7, s=2. fallfac(n, m) := A008279(n, m) (falling factorials triangle).

cp's OEIS Frontend

A090439 Alternating row sums of array A090438 ((4,2)-Stirling2).

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A091032 Second column (k=3) of array A090438 ((4,2)-Stirling2) divided by 8.

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A091033 Third column (k=4) of array A090438 ((4,2)-Stirling2).

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A091034 Fourth column (k=5) of array A090438 ((4,2)-Stirling2) divided by 24.

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A091035 Fifth column (k=6) of array A090438 ((4,2)-Stirling2).

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A091036 Sixth column (k=7) of array A090438 ((4,2)-Stirling2) divided by 48=4!*2.

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A002674 a(n) = (2n)!/2.

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A091534 Generalized Stirling2 array (5,2).

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A091746 Generalized Stirling2 array (6,2).