A010790 -id:A010790 - cp's OEIS frontend

A288035 Number of (undirected) paths in the complete bipartite graph K_n,n.

1, 12, 135, 2224, 55725, 2006316, 98309827, 6291829440, 509638185369, 50963818537900, 6166622043087231, 887993574204562992, 150070914040571147845, 29413899151951944980364, 6618127309189187620585275, 1694240591152432030869834496, 489635530843052856921382174257

Offset: 1

Eric W. Weisstein, Jun 04 2017

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..50
Eric Weisstein's World of Mathematics, Complete Bipartite Graph
Eric Weisstein's World of Mathematics, Graph Path

Main diagonal of A307027 and A360850.

Cf. A070968, A048617, A010790.

Table[Sum[(n!)^2/((n - Ceiling[k/2])! (n - Floor[k/2])!), {k, 2, 2 n}], {n, 20}] (* Eric W. Weisstein, Jun 13 2017 *)
Table[n!^2 (BesselI[0, 2] + BesselI[1, 2] - HypergeometricPFQRegularized[{1}, {1 + n, 1 + n}, 1]) - n HypergeometricPFQ[{1}, {n, 1 + n}, 1], {n, 20}] // FunctionExpand (* Eric W. Weisstein, Jun 13 2017 *)

a(n) = sum(k=2, 2*n, n!^2/((n-(k+1)\2)!*(n-k\2)!)); \\ Andrew Howroyd, Jun 10 2017

a(n) = n!^2*sum(k=0, n-1, (1 + k)/(k!)^2) \\ Andrew Howroyd, Feb 24 2023

a(n) = Sum_{k=2..2*n} n!^2/((n-ceiling(k/2))!*(n-floor(k/2))!). - Andrew Howroyd, Jun 10 2017

a(n) = n!^2 * Sum_{k=0..n-1} (1 + k)/(k!^2). - Andrew Howroyd, Feb 24 2023

Terms a(8) and beyond from Andrew Howroyd, Jun 10 2017

A351430 a(n) is the number of reduced stable marriage problem instances of order 4 that generate n possible stable matchings.

Original entry on oeis.org

457411536, 249495038, 50719534, 5983183, 774164, 24157, 4038, 253, 0, 1

Offset: 1

Dan Eilers, Feb 11 2022

A344667(10) is reduced from 144 to 1, demonstrating that it is a unique maximal instance up to relabeling of the participants.

Cf. A344667, A351409, A010790.

a(n) = A344667(n)/A010790(3) as described in A351409.

A090443 a(n) = (n+2)! * (n+1)! * n! / 2.

Original entry on oeis.org

1, 6, 144, 8640, 1036800, 217728000, 73156608000, 36870930432000, 26547069911040000, 26281599211929600000, 34691710959747072000000, 59530976006925975552000000, 130015651599126330605568000000, 354942728865614882553200640000000, 1192607568988466005378754150400000000

Offset: 0

Wolfdieter Lang, Dec 23 2003

Harvey P. Dale, Table of n, a(n) for n = 0..180
J. Agapito, On symmetric polynomials with only real zeros and nonnegative gamma-vectors, Linear Algebra and its Applications, Volume 451, 15 June 2014, Pages 260-289.
César Aguilera, On Zeta Functions and Tetrahedral Numbers, hal-02297262 (2019) [math.NT].

Cf. A010790, A090444, A224900, A172492.

a:=n->mul(j^3-j, j=2..n): seq(a(n), n=1..13); # Zerinvary Lajos, May 08 2008

(Times@@#)/2&/@Partition[Range[0,20]!,3,1] (* Harvey P. Dale, Dec 03 2017 *)

from math import factorial
def A090443(n): return factorial(n)**3*(n+2)*(n+1)**2>>1 # Chai Wah Wu, Apr 22 2024

Fourth column (m=3) of triangle A090441.
From Karol A. Penson Jul 25 2013: (Start)
G.f. of hypergeometric type:
  Sum_{n>=0} a(n)*z^n/(n!)^3 = (1+2*z)/(1-z)^4;
integral representation as n-th moment of a positive function w(x) on a positive halfaxis (solution of the Stieltjes moment problem), in Maple notation:
a(n) = int(x^n*w(x),x=0..infinity), n>=0 where w(x)=MeijerG([[],[]],[[2,1,0]],[]],x)/2, w(0)=1/2, limit(w(x),x=infinity)=0. w(x) is monotonically decreasing over (0,infinity). The Meijer G function above cannot be represented by any other known special function.
This solution of the Stieltjes moment problem is not unique.
Asymptotics: a(n)->(1/32)*Pi^(3/2)*sqrt(2)*(32*n^2+136*n+193)*exp(-3*n)*(n)^(5/2+3*n), for n->infinity. (End)

A084915 a(n) = (n!)^2*n.

Original entry on oeis.org

0, 1, 8, 108, 2304, 72000, 3110400, 177811200, 13005619200, 1185137049600, 131681894400000, 17526860144640000, 2753310393630720000, 504085244567224320000

Offset: 0

Jon Perry, Jul 14 2003

nonn

Used to prove that Sum_{n>=1} 1/A002378(n) = 1. Examining Sum_{n=1..k} 1/A002378(n) gives 1/2, 1/2 + 1/6, 1/2 + 1/6 + 1/12. Simplifying gives 1/2, 8/12, 108/144, where the numerators are this sequence and the denominators are A010790. Therefore we have k!^2*k/k!(k+1)! = k*k!/(k+1)! = k/(k+1), which tends to 1 as k tends to infinity.

			a(3) = 3!^2*3 = 36*3 = 108.

Cf. A002378, A010790, A071951.

PARI
```
for(n=1,50,print1(n!^2*n","))
```

a(n) = n!*(n+1)! - n!^2.

a(n) = det(PS(i+2,j+1), 1 <= i,j <= n-1), where PS(n,k) are Legendre-Stirling numbers of the second kind (A071951) and n > 0. [Mircea Merca, Apr 06 2013]

A130033 Fourth (m=3) column sequence of triangle A129467.

Original entry on oeis.org

1, -20, 508, -17544, 808848, -48405888, 3663035136, -342678781440, 38879803008000, -5263815891456000, 838682139211776000, -155393459730173952000, 33136711787903754240000, -8059211591488628981760000, 2217755736675770074398720000

Offset: 0

Wolfdieter Lang, May 04 2007

See the M. Bruschi et al. reference given in A129467.

			a(3)=-det([20,1,0],[292,40,1],[3824,1092,70])=-17544. [_Mircea Merca_, Apr 06 2013]

G. C. Greubel, Table of n, a(n) for n = 0..250

Cf. A010790 (m=1 column unsigned), A084915 (m=2 column unsigned).

Cf. A129467.

h:= func< n,k | (&+[1/j^k : j in [1..n]]) >;
A130033:= func< n | (-1)^n*(Factorial(n+2))^2*(2*(n+2) - (n+3)*h(n+2,2)) >;
[A130033(n): n in [0..30]]; // G. C. Greubel, Feb 10 2024

A130033[n_]:= (-1)^n*((n+2)!)^2*(2*(n+2) -(n+3)*HarmonicNumber[n+2,2]);
Table[A130033[n], {n,0,30}] (* G. C. Greubel, Feb 10 2024 *)

def A130033(n): return (-1)^n*(factorial(n+2))^2*(2*(n+2) - (n+3)*(zeta(2) - psi(1,n+3)))
[A130033(n) for n in range(31)] # G. C. Greubel, Feb 10 2024

a(n) = A129467(n+3,3),n>=0.
a(n) = (-1)^n*det(PS(i+3,j+2), 1 <= i,j <= n), where PS(n,k) are Legendre-Stirling numbers of the second kind (A071951). - Mircea Merca, Apr 06 2013
a(n) = (-1)^n * ((n+2)!)^2 * (2*(n+2) - (n+3)*h(n+2, 2)), where h(n,k) = Sum_{j=1..n} 1/j^k is the generalized harmonic number. - G. C. Greubel, Feb 10 2024

A130559 Coefficients of the v=n member of a family of certain orthogonal polynomials with Diophantine properties.

Original entry on oeis.org

1, -2, 1, 12, -8, 1, -144, 108, -20, 1, 2880, -2304, 508, -40, 1, -86400, 72000, -17544, 1708, -70, 1, 3628800, -3110400, 808848, -89280, 4648, -112, 1, -203212800, 177811200, -48405888, 5808528, -349568, 10920, -168, 1, 14631321600, -13005619200, 3663035136, -466619904

Offset: 0

Wolfdieter Lang, Jul 13 2007

For v>=1 the orthogonal polynomials pt(n,v,x) have only integer zeros k*(k+1), k=1..n These integer zeros are from 2*A000217.
Coefficients of pt(n,v=n,x) (in the quoted Bruschi et al. paper {\tilde p}^{(\nu)}_n(x) of eqs. (20) and (24a),(24b)) in increasing powers of x.
The v-family pt(n,v,x) consists of characteristic polynomials of the tridiagonal M x M matrix Vt=Vt(M,v) with entries Vt_{m,n} given by 2*m*(v+1-m) if n=m, m=1,...,M; -m*(v+1-m) if n=m-1, m=2,...,M; -m*(v+1-m) if n=m+1, m=1..M-1 and 0 else. pt(n,v,x):=det(x*I_n-Vt(n,v) with the n dimensional unit matrix I_n.
pt(n,v=n,x) has, for every n>=1, the n integer zeros 2,6,12,...,n*(n+1). pt(2,2,x) has therefore only the integer zeros 2 and 6. 12= 2*6 = det(Vt(2,2))=16-4.
This triangle coincides with triangle A129467 without row n=0 and column m=0, taking as offset again [0,0].
Column sequences give for m=0..2: A010790(n-1)*(-1)^(n-1), A084915(n+1)*(-1)^n, A130033.

			n=2: [12,-8,1] stands for pt(2,2,x) = 12-8*x+x^2 = (x-2)*(x-6) with the integer zeros 2*1 and 2*3.
Triangle begins:
  [1];
  [-2,1];
  [12,-8,1];
  [-144,108,-20,1];
  [2880,-2304,508,-40,1];
  ...

M. Bruschi, F. Calogero and R. Droghei, Proof of certain Diophantine conjectures and identification of remarkable classes of orthogonal polynomials, J. Physics A, 40(2007), pp. 3815-3829.
M. W. Coffey and M. C. Lettington, On Fibonacci Polynomial Expressions for Sums of mth Powers, their implications for Faulhaber's Formula and some Theorems of Fermat, arXiv:1510.05402 [math.NT], 2015.
Wolfdieter Lang, First 10 rows and more.

Row sums give A130031(n+1), n>=0. Unsigned row sums give A130032(n+1), n>=1.

Cf. A130182 (v=1 member).

a(n,m) = [x^m]pt(n,n,x), n>=0, with the three term recurrence for orthogonal polynomial systems of the form pt(n,v,x) = (x + 2*n*(n-1-v))*pt(n-1,v,x) - (n-1)*n*(n-1-v)*(n-2-v)*pt(n-2,v,x), n>=1; pt(-1,v,x) = 0 and pt(0,v,x) = 1. Start with v = n.

A134287 Fifth column of triangle A103371 (without leading zeros).

Original entry on oeis.org

1, 30, 315, 1960, 8820, 31752, 97020, 261360, 637065, 1431430, 3006003, 5962320, 11262160, 20391840, 35581680, 60093504, 98590905, 157608990, 246142435, 376372920, 564559380, 832117000, 1206913500, 1724814000, 2431508625

Offset: 0

Wolfdieter Lang, Nov 13 2007

Kekulé numbers for certain benzenoids.
a(n) = K(L(n))*K(O(2,4,n)) with the Cyvin and Gutman Kekulé number notation. See p. 62 for the L(n) structure with K(L(n))=n+1 and p. 105 (i) for the O(k,m,n) structure and its Kekulé number. This corresponds to an essentially disconnected 7-tier benzenoid structure similar to the 6-tier structure shown on p. 230, nr. 23 (see A108647).
a(n-5), n >= 5, is the number of ways to put n identical objects into m=5 of altogether n distinguishable boxes (n-5 boxes stay empty).

			a(2)=315 because n=7 identical balls can be put into m=5 of n=7 distinguishable boxes in binomial(7,5)*(5!/(4!*1!)+ 5!/(3!*2!)) = 21*(5+10) = 315 ways. The m=5 part partitions of 7, namely (1^4,3) and (1^3,2^2) specify the filling of each of the 21 possible five box choices. - _Wolfdieter Lang_, Nov 13 2007

S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Cf. A108647 (fourth column of triangle A103371).

a134287 = flip a103371 4 . (+ 4)  -- Reinhard Zumkeller, Apr 04 2014

[5*Binomial(n+5, 5)^2/(n+5): n in [0..30]]; // G. C. Greubel, Oct 28 2022

seq(binomial(n+4,4)^2*(n+5)/5, n=0..24); # Peter Luschny, Jan 13 2014

CoefficientList[Series[(1 + 20 x + 60 x^2 + 40 x^3 + 5 x^4)/(1 - x)^10, {x, 0, 24}], x]

5*binomial(n+5,5)^2/(n+5) $ n = 0..35; // Zerinvary Lajos, May 09 2008

a(n) = 5*binomial(n+5, 5)^2/(n+5); \\ Michel Marcus, Jan 07 2014

[5*binomial(n+5,5)^2/(n+5) for n in range(31)] # G. C. Greubel, Oct 28 2022

a(n) = A103371(n+4,4), n >= 0.
a(n) = ((n+1)*(n+2)*(n+3)*(n+4))^2*(n+5)/2880, n >= 0. 2880 = 4!*5! = A010790(4).
G.f.: (1+20*x+60*x^2+40*x^3+5*x^4)/(1-x)^10. Numerator polynomial from fifth row of triangle A132813.
a(n) = 5*C(n+5,5)^2/(n+5), n >= 0. - Zerinvary Lajos, May 09 2008
a(n) = (C(n+6,6)*C(n+5,4)+5*C(n+5,6)*C(n+5,4))/(n+5). - Gary Detlefs, Jan 06 2014
From Amiram Eldar, May 31 2022: (Start)
Sum_{n>=0} 1/a(n) = 350*Pi^2/3 - 13805/12.
Sum_{n>=0} (-1)^n/a(n) = 5*Pi^2 + 640*log(2)/3 - 785/4. (End)
E.g.f.: (2880 + 83520*x + 368640*x^2 + 529920*x^3 + 330120*x^4 + 102024*x^5 + 16616*x^6 + 1432*x^7 + 61*x^8 + x^9)*exp(x)/2880. - G. C. Greubel, Oct 28 2022

A290057 Number T(n,k) of X-rays of n X n binary matrices with exactly k ones; triangle T(n,k), n>=0, 0<=k<=n^2, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 3, 4, 3, 1, 1, 5, 13, 23, 30, 30, 23, 13, 5, 1, 1, 7, 26, 68, 139, 234, 334, 411, 440, 411, 334, 234, 139, 68, 26, 7, 1, 1, 9, 43, 145, 386, 860, 1660, 2838, 4362, 6090, 7779, 9135, 9892, 9892, 9135, 7779, 6090, 4362, 2838, 1660, 860, 386, 145, 43, 9, 1

Offset: 0

Alois P. Heinz, Jul 19 2017

The X-ray of a matrix is defined as the sequence of antidiagonal sums.

T(n,k) is defined for all n,k >= 0. The triangle contains only the positive terms. T(n,k) = 0 for k>n^2.

			Triangle T(n,k) begins:
  1;
  1, 1;
  1, 3,  4,  3,   1;
  1, 5, 13, 23,  30,  30,  23,  13,   5,   1;
  1, 7, 26, 68, 139, 234, 334, 411, 440, 411, 334, 234, 139, 68, 26, 7, 1;
  ...

Alois P. Heinz, Rows n = 0..40, flattened
C. Bebeacua, T. Mansour, A. Postnikov and S. Severini, On the X-rays of permutations, arXiv:math/0506334 [math.CO], 2005.
Index entries for sequences related to binary matrices

Columns k=0-2 give: A000012, A004273, A091823(n-1) for n>1.
Main diagonal gives A290052.
Row sums give A010790.
Cf. A000290, A290058.

b:= proc(n, i, t) option remember; (m-> `if`(n>m, 0, `if`(n=m, 1,
      add(b(n-j, i-t, 1-t), j=0..min(i, n)))))(i*(i+1-t))
    end:
T:= (n, k)-> b(k, n, 1):
seq(seq(T(n, k), k=0..n^2), n=0..7);

b[n_,i_,t_]:= b[n, i, t] = Function[{m, jm}, If[n>m, 0, If[n==m, 1, Sum[b[n-j, i-t, 1-t], {j, 0, jm}]]]][i*(i+1-t), Min[i, n]]; T[n_, k_]:= b[k, n, 1]; Table[T[n, k], {n, 0, 7}, {k, 0, n^2}] // Flatten (* Jean-François Alcover, Aug 09 2017, translated from Maple *)

T(n,floor(n^2/2)) = A290058(n).

T(n,k) = T(n,n^2-k).

A290134 Number of unique X-rays of n X n binary matrices with exactly floor(n^2/2) ones.

Original entry on oeis.org

1, 1, 2, 5, 14, 42, 130, 415, 1368, 4603, 15788, 54863, 193112, 686049, 2459942, 8881931, 32292148, 118038070, 433790834, 1601042055, 5934546466, 22074679425, 82399006636, 308471888767, 1158175006638, 4359154749776, 16447468190380, 62188658733901

Offset: 0

Alois P. Heinz, Jul 20 2017

nonn

The X-ray of a matrix is defined as the sequence of antidiagonal sums.
A unique X-ray allows reconstruction of the binary matrix.
The number of unique X-rays of all n X n binary matrices is A081294(n).
The number of all X-rays of n X n binary matrices is A010790(n).

			a(3) = 5: 00301, 02020, 10021, 10300, 12001.
a(4) = 14: 0004301, 0030320, 0034001, 0200321, 0204020, 0230021, 0230300, 1004021, 1004300, 1030301, 1034000, 1200320, 1204001, 1230020.

Alois P. Heinz, Table of n, a(n) for n = 0..750
C. Bebeacua, T. Mansour, A. Postnikov and S. Severini, On the X-rays of permutations, arXiv:math/0506334 [math.CO], 2005.
Index entries for sequences related to binary matrices

Cf. A010790, A081294, A290133, A290058.

b:= proc(n, i, t) option remember; (m-> `if`(n>m, 0, `if`(n=m, 1,
      b(n, i-t, 1-t)+`if`(i>n, 0, b(n-i, i-t, 1-t)))))(i*(i+1-t))
    end:
a:= n-> b(iquo(n^2, 2), n, 1):
seq(a(n), n=0..40);

b[n_, i_, t_] := b[n, i, t] = Function[m, If[n > m, 0, If[n == m, 1, b[n, i-t, 1-t] + If[i > n, 0, b[n - i, i - t, 1 - t]]]]][i*(i + 1 - t)];
a[n_] := b[Quotient[n^2, 2], n, 1];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Nov 06 2017, after Alois P. Heinz *)

a(n) ~ sqrt(3) * 2^(2*n-1) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Jul 22 2017

A348607 Decimal expansion of BesselJ(1,2).

Original entry on oeis.org

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

Offset: 0

Dumitru Damian, Oct 25 2021

			0.5767248077568733872...

Cf. A010790, A054687, A058798, A058797, A133920, A245308, A296168, A301484, A318224.

Bessel function values: A334380 (J(0,1)), A091681 (J(0,2)), A334383 (J(0,sqrt(2))), this sequence (J(1,2)), A197036 (I(0,1)), A070910 (I(0,2)), A334381 (I(0,sqrt(2))), A096789 (I(1,2)).

RealDigits[BesselJ[1, 2], 10, 100][[1]] (* Amiram Eldar, Oct 25 2021 *)

besselj(1, 2) \\ Michel Marcus, Oct 25 2021

Sage
```
bessel_J(1, 2).n(digits=100)
```

Equals Sum_{k>=0} (-1)^k/(k!*(k+1)!).

cp's OEIS Frontend

A288035 Number of (undirected) paths in the complete bipartite graph K_n,n.

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A351430 a(n) is the number of reduced stable marriage problem instances of order 4 that generate n possible stable matchings.

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A090443 a(n) = (n+2)! * (n+1)! * n! / 2.

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A084915 a(n) = (n!)^2*n.

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A130033 Fourth (m=3) column sequence of triangle A129467.

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A130559 Coefficients of the v=n member of a family of certain orthogonal polynomials with Diophantine properties.

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A134287 Fifth column of triangle A103371 (without leading zeros).

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