A343832 -id:A343832 - cp's OEIS frontend

A082545 a(n) = (2n)! Sum_{k=0..n} binomial(n,k)/(n+k)!.

1, 3, 21, 229, 3393, 63591, 1442173, 38398641, 1174226049, 40558249963, 1561734494661, 66335687785533, 3081211226192641, 155369391396527439, 8452596370942940973, 493494408990278911561, 30777323181433121541633, 2042075395611656190239571

Offset: 0

Vladeta Jovovic, May 11 2003

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..365

Cf. A006902, A343832.

[Factorial(n)*Evaluate(LaguerrePolynomial(n, n), -1): n in [0..40]]; // G. C. Greubel, Aug 11 2022

a:= n-> simplify(n!*LaguerreL(n$2, -1)):
seq(a(n), n=0..20);  # Alois P. Heinz, Jul 27 2017

Table[n!*LaguerreL[n, n, -1], {n,0,17}] (* Jean-François Alcover, Jun 04 2019 *)

a(n) = sum(k=0, n, k!*binomial(n, k)*binomial(2*n, k)); \\ Seiichi Manyama, May 01 2021

a(n) = n!*pollaguerre(n, n, -1); \\ Seiichi Manyama, May 01 2021

[factorial(n)*gen_laguerre(n, n, -1) for n in (0..40)] # G. C. Greubel, Aug 11 2022

a(n) = n!*LaguerreL(n, n, -1).
n*a(n) + (n^3-5*n^2-n+2)*a(n-1) - 2*(n+1)*(2*n-3)*(n-1)^2*a(n-2) = 0. - Vladeta Jovovic, Jul 16 2004
E.g.f.: exp((-2*x+1-(1-4*x)^(1/2))/(2*x))/(1-4*x)^(1/2). - Mark van Hoeij, Oct 31 2011
a(n) ~ n^n*2^(2*n+1/2)/exp(n-1). - Vaclav Kotesovec, Sep 27 2012
a(n) = n!*binomial(2*n,n)*hypergeom([-n], [1+n], -1). - Peter Luschny, May 04 2017
a(n) = n! * [x^n] exp(x/(1 - x))/(1 - x)^(n+1). - Ilya Gutkovskiy, Nov 21 2017

A152059 a(n) is the number of ways 2n-1 seats can be occupied by at most n people for n>=1, with a(0)=1.

Original entry on oeis.org

1, 2, 13, 136, 1961, 36046, 805597, 21204548, 642451441, 22021483546, 842527453421, 35591363004352, 1645373927307673, 82625931422081126, 4478815087922020861, 260648364396903639676, 16208855884741850686817

Offset: 0

Paul D. Hanna, Nov 22 2008

nonn

Let A(x) be the e.g.f. of this sequence, and B(x) be the e.g.f. of A082545, then B(x)/A(x) = C(x) where C(x) = 1 + x*C(x)^2 is the Catalan function (A000108). This follows from the fact that this sequence and A082545 form adjacent semi-diagonals of table A088699. - Paul D. Hanna, Aug 16 2022

Seiichi Manyama, Table of n, a(n) for n = 0..366
Wikipedia, Laguerre polynomials
Index entries for sequences related to Laguerre polynomials

Cf. A082545, A086885, A343832, A088699, A000108.

[Factorial(n)*Evaluate(LaguerrePolynomial(n, n-1), -1): n in [0..40]]; // G. C. Greubel, Aug 11 2022

Table[(-1)^n * HypergeometricU[-n, n, -1], {n, 0, 20}] (* Vaclav Kotesovec, Oct 02 2017 *)

a(n)=sum(k=0,n,k!*binomial(2*n-1, k)*binomial(n, k))

a(n) = n!*pollaguerre(n, n-1, -1); \\ Seiichi Manyama, May 01 2021

[factorial(n)*gen_laguerre(n, n-1, -1) for n in (0..40)] # G. C. Greubel, Aug 11 2022

a(n) = Sum_{k=0..n} k! * C(2*n-1,k) * C(n,k).
Central terms of triangle A086885 (after initial term).
a(n) = n! * [x^n] exp(x/(1 - x))/(1 - x)^n. - Ilya Gutkovskiy, Oct 02 2017
a(n) ~ 2^(2*n - 1/2) * n^n / exp(n-1). - Vaclav Kotesovec, Oct 02 2017
a(n) = n! * pollaguerre(n, n-1, -1). - Seiichi Manyama, May 01 2021
From Paul D. Hanna, Aug 16 2022: (Start)
E.g.f.: exp( (1-2*x - sqrt(1-4*x))/(2*x) ) / ((sqrt(1-4*x) - (1-4*x))/(2*x)), derived from the e.g.f for A082545 given by Mark van Hoeij.
E.g.f.: exp(C(x) - 1) / (2 - C(x)), where C(x) = (1 - sqrt(1-4*x))/(2*x) is the Catalan function (A000108). (End)

A176120 Triangle read by rows: Sum_{j=0..k} binomial(n, j)binomial(k, j)j!.

Original entry on oeis.org

1, 1, 2, 1, 3, 7, 1, 4, 13, 34, 1, 5, 21, 73, 209, 1, 6, 31, 136, 501, 1546, 1, 7, 43, 229, 1045, 4051, 13327, 1, 8, 57, 358, 1961, 9276, 37633, 130922, 1, 9, 73, 529, 3393, 19081, 93289, 394353, 1441729, 1, 10, 91, 748, 5509, 36046, 207775, 1047376, 4596553, 17572114

Offset: 0

Roger L. Bagula, Apr 09 2010

The number of ways of placing any number k = 0, 1, ..., min(n,m) of non-attacking rooks on an n X m chessboard. - R. J. Mathar, Dec 19 2014

			Triangle begins
  1;
  1,  2;
  1,  3,   7;
  1,  4,  13,   34;
  1,  5,  21,   73,  209;
  1,  6,  31,  136,  501,  1546;
  1,  7,  43,  229, 1045,  4051,  13327;
  1,  8,  57,  358, 1961,  9276,  37633,  130922;
  1,  9,  73,  529, 3393, 19081,  93289,  394353,  1441729;
  1, 10,  91,  748, 5509, 36046, 207775, 1047376,  4596553, 17572114;
  1, 11, 111, 1021, 8501, 63591, 424051, 2501801, 12975561, 58941091, 234662231;

O. Ganyushkin and V. Mazorchuk, Classical Finite Transformation Semigroups, Springer, 2009, page 46.

Seiichi Manyama, Rows n = 0..139, flattened
Wikipedia, Rook polynomial

Cf. A086885 (table without column 0), A129833 (row sums).

Cf. A000262, A002061, A002720, A082545, A135859, A343832.

A176120:=func< n,k| (&+[Factorial(j)*Binomial(n,j)*Binomial(k,j): j in [0..k]]) >;
[A176120(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Aug 11 2022

A176120 := proc(i,j)
        add(binomial(i,k)*binomial(j,k)*k!,k=0..j) ;
end proc: # R. J. Mathar, Jul 28 2016

T[n_, m_]:= T[n,m]= Sum[Binomial[n, k]*Binomial[m, k]*k!, {k, 0, m}];
Table[T[n, m], {n,0,12}, {m,0,n}]//Flatten

def A176120(n,k): return sum(factorial(j)*binomial(n,j)*binomial(k,j) for j in (0..k))
flatten([[A176120(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Aug 11 2022

Sum_{k=0..n} T(n, k) = A129833(n).
T(n,m) = A088699(n, m). - Peter Bala, Aug 26 2013
T(n,m) = A086885(n, m). - R. J. Mathar, Dec 19 2014
From G. C. Greubel, Aug 11 2022: (Start)
T(n, k) = Hypergeometric2F1([-n, -k], [], 1).
T(2*n, n) = A082545(n).
T(2*n+1, n) = A343832(n).
T(n, n) = A002720(n).
T(n, n-1) = A000262(n), n >= 1.
T(n, 1) = A000027(n+1).
T(n, 2) = A002061(n+1).
T(n, 3) = A135859(n+1). (End)

A343830 a(n) = numerator of (1/e) * Sum_{a_1>=1, a_2>=1, ... , a_n>=1} a_1 * a_2 * ... * a_n / (a_1 + a_2 + ... + a_n)!.

Original entry on oeis.org

1, 2, 31, 179, 787, 6631, 2456299, 33235913, 158433901, 17980176031, 2794938616471, 8546650588601, 5595650767265101, 35480190026972501, 15523069639558351459, 455264603021602214213, 57023540590242398853649, 949437664962426221725789, 5469912218467062529961407

Offset: 1

Seiichi Manyama, Apr 30 2021

			1, 2/3, 31/120, 179/2520, 787/51840, 6631/2494800, 2456299/6227020800, ...

O. Furdui, Limits, Series and Fractional Part Integrals. Problems in Mathematical Analysis, Springer, New York, 2013. See Problem 3.114 and 3.118.

Seiichi Manyama, Table of n, a(n) for n = 1..367
Math StackExchange, Compute S_n = Sum_{a_1 a_2 ... a_n >=1} a_1 a_2 ... a_n/(a_1+a_2+...+a_n)!

Cf. A343831 (denominator), A343832.

a[n_] := Numerator @ Sum[Binomial[n - 1, k]/(k + n)!, {k, 0, n - 1}]; Array[a, 20] (* Amiram Eldar, May 01 2021 *)

a(n) = numerator(sum(j=0, n, (-1)^(n+j-1)*binomial(n, j)*sum(k=0, n+j-1, (-1)^k/k!)));

a(n) = numerator(sum(k=0, n-1, binomial(n-1, k)/(k+n)!));

b(n) = (1/e) * Sum_{a_1>=1, a_2>=1, ... , a_n>=1} a_1 * a_2 * ... * a_n / (a_1 + a_2 + ... + a_n)! = Sum_{j=0..n} (-1)^(n+j-1) * binomial(n,j) * Sum_{k=0..n+j-1} (-1)^k/k! = Sum_{k=0..n-1} binomial(n-1,k)/(k+n)!.

a(n) = numerator of b(n).

A343831 a(n) = denominator of (1/e) * Sum_{a_1>=1, a_2>=1, ... , a_n>=1} a_1 * a_2 * ... * a_n / (a_1 + a_2 + ... + a_n)!.

Original entry on oeis.org

1, 3, 120, 2520, 51840, 2494800, 6227020800, 653837184000, 27360571392000, 30411275102208000, 51090942171709440000, 1846572624206069760000, 15511210043330985984000000, 1361108681302294020096000000, 8841761993739701954543616000000

Offset: 1

Seiichi Manyama, Apr 30 2021

			1, 2/3, 31/120, 179/2520, 787/51840, 6631/2494800, 2456299/6227020800, ...

O. Furdui, Limits, Series and Fractional Part Integrals. Problems in Mathematical Analysis, Springer, New York, 2013. See Problem 3.114 and 3.118.

Seiichi Manyama, Table of n, a(n) for n = 1..225
Math StackExchange, Compute S_n = Sum_{a_1 a_2 ... a_n >=1} a_1 a_2 ... a_n/(a_1+a_2+...+a_n)!

Cf. A343830 (numerator), A343832.

a[n_] := Denominator @ Sum[Binomial[n - 1, k]/(k + n)!, {k, 0, n - 1}]; Array[a, 20] (* Amiram Eldar, May 01 2021 *)

a(n) = denominator(sum(j=0, n, (-1)^(n+j-1)*binomial(n, j)*sum(k=0, n+j-1, (-1)^k/k!)));

a(n) = denominator(sum(k=0, n-1, binomial(n-1, k)/(k+n)!));

b(n) = (1/e) * Sum_{a_1>=1, a_2>=1, ... , a_n>=1} a_1 * a_2 * ... * a_n / (a_1 + a_2 + ... + a_n)! = Sum_{j=0..n} (-1)^(n+j-1) * binomial(n,j) * Sum_{k=0..n+j-1} (-1)^k/k! = Sum_{k=0..n-1} binomial(n-1,k)/(k+n)!.

a(n) = denominator of b(n).

A343896 a(n) = Sum_{k=0..n} (-1)^(n-k) * k! * binomial(n,k) * binomial(2*n+1,k).

Original entry on oeis.org

1, 2, 11, 104, 1405, 24694, 534223, 13719404, 407730041, 13760958410, 519827337331, 21726980525392, 995403499490101, 49600090942276094, 2670566242480261175, 154500457959360271124, 9557826199486960327153, 629586464929967678553874, 43994787057844036765113691

Offset: 0

Seiichi Manyama, May 03 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..366
Wikipedia, Laguerre polynomials
Index entries for sequences related to Laguerre polynomials

Cf. A006902, A343832, A343890.

a[n_] := n!*LaguerreL[n, n + 1, 1]; Array[a, 19, 0] (* Amiram Eldar, May 11 2021 *)

a(n) = sum(k=0, n, (-1)^(n-k)*k!*binomial(n, k)*binomial(2*n+1, k));

a(n) = (2*n+1)!*sum(k=0, n, (-1)^k*binomial(n, k)/(k+n+1)!);

a(n) = n!*sum(k=0, n, (-1)^(n-k)*binomial(2*n+1, k)/(n-k)!);

PARI
```
a(n) = n!*pollaguerre(n, n+1, 1);
```

a(n) = (2*n+1)! * Sum_{k=0..n} (-1)^k * binomial(n,k)/(k+n+1)!.
a(n) = n! * Sum_{k=0..n} (-1)^(n-k) * binomial(2*n+1,k)/(n-k)!.
a(n) = n! * LaguerreL(n, n+1, 1).
a(n) = n! * [x^n] exp(-x/(1 - x))/(1 - x)^(n+2).
a(n) ~ 2^(2*n + 3/2) * n^n / exp(n+1). - Vaclav Kotesovec, May 03 2021

A380491 a(n) = n! * Sum_{k=0..n} binomial(2*n-3,k)/(n-k)!.

Original entry on oeis.org

1, 0, 3, 34, 501, 9276, 207775, 5470158, 165625929, 5671386136, 216730118331, 9144481575450, 422249317829053, 21180324426577044, 1146880568461500951, 66677192513929212166, 4142571510546929867025, 273910161452560881843888, 19204878684852222745880179

Offset: 0

Seiichi Manyama, Jan 25 2025

nonn

Cf. A082545, A152059, A251568, A293985, A343832, A380492, A380493.

PARI
```
a(n) = n!*pollaguerre(n, n-3, -1);
```

a(n) = n! * LaguerreL(n, n-3, -1).

a(n) = n! * [x^n] exp(x/(1 - x))/(1 - x)^(n-2).

A380492 a(n) = n! * Sum_{k=0..n} binomial(2*n-2,k)/(n-k)!.

Original entry on oeis.org

1, 1, 7, 73, 1045, 19081, 424051, 11109337, 335262313, 11453449105, 436944953791, 18412283563081, 849345673881277, 42570185481576793, 2303643608370636715, 133859418832759525081, 8312945340897388101841, 549460711493172343519777, 38513032385247860120975863

Offset: 0

Seiichi Manyama, Jan 25 2025

nonn

Cf. A082545, A152059, A251568, A293985, A343832, A380491, A380493.

PARI
```
a(n) = n!*pollaguerre(n, n-2, -1);
```

a(n) = n! * LaguerreL(n, n-2, -1).

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

A380493 a(n) = n! * Sum_{k=0..n} binomial(2*n+3,k)/(n-k)!.

Original entry on oeis.org

1, 6, 57, 748, 12585, 259026, 6315001, 178134552, 5711078673, 205209960670, 8171229107481, 357235056697476, 17014791129640057, 877089297426429738, 48657292133825026905, 2890717184573264397616, 183125115830192864360481, 12323226433255671469949622

Offset: 0

Seiichi Manyama, Jan 25 2025

nonn

Cf. A082545, A152059, A251568, A293985, A343832, A380491, A380492.

PARI
```
a(n) = n!*pollaguerre(n, n+3, -1);
```

a(n) = n! * LaguerreL(n, n+3, -1).

a(n) = n! * [x^n] exp(x/(1 - x))/(1 - x)^(n+4).

A343890 Coefficient triangle of generalized Laguerre polynomials n!*L(n,n+1,x) (rising powers of x).

Original entry on oeis.org

1, 3, -1, 20, -10, 1, 210, -126, 21, -1, 3024, -2016, 432, -36, 1, 55440, -39600, 9900, -1100, 55, -1, 1235520, -926640, 257400, -34320, 2340, -78, 1, 32432400, -25225200, 7567560, -1146600, 95550, -4410, 105, -1, 980179200, -784143360, 249500160, -41583360, 3998400, -228480, 7616, -136, 1

Offset: 0

Seiichi Manyama, May 03 2021

			The triangle begins:
        1;
        3,      -1;
       20,     -10,      1;
      210,    -126,     21,     -1;
     3024,   -2016,    432,    -36,    1;
    55440,  -39600,   9900,  -1100,   55,  -1;
  1235520, -926640, 257400, -34320, 2340, -78, 1;

Row sums (signed) give A343896, row sums (unsigned) give A343832.

Cf. A343861.

T[n_, k_] := (-1)^k * (2*n + 1)! * Binomial[n, k]/(k + n + 1)!; Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Amiram Eldar, May 03 2021 *)

T(n, k) = (-1)^k*(2*n+1)!*binomial(n,k)/(k+n+1)!;

row(n) = Vecrev(n!*pollaguerre(n, n+1));

T(n, k) = (-1)^k * n! * binomial(2*n+1,n-k)/k! = (-1)^k * (2*n+1)! * binomial(n,k)/(k+n+1)!.

cp's OEIS Frontend

A082545 a(n) = (2*n)! * Sum_{k=0..n} binomial(n,k)/(n+k)!.

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A152059 a(n) is the number of ways 2n-1 seats can be occupied by at most n people for n>=1, with a(0)=1.

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A176120 Triangle read by rows: Sum_{j=0..k} binomial(n, j)*binomial(k, j)*j!.

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A343830 a(n) = numerator of (1/e) * Sum_{a_1>=1, a_2>=1, ... , a_n>=1} a_1 * a_2 * ... * a_n / (a_1 + a_2 + ... + a_n)!.

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A343831 a(n) = denominator of (1/e) * Sum_{a_1>=1, a_2>=1, ... , a_n>=1} a_1 * a_2 * ... * a_n / (a_1 + a_2 + ... + a_n)!.

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A343896 a(n) = Sum_{k=0..n} (-1)^(n-k) * k! * binomial(n,k) * binomial(2*n+1,k).

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A082545 a(n) = (2n)! Sum_{k=0..n} binomial(n,k)/(n+k)!.

A176120 Triangle read by rows: Sum_{j=0..k} binomial(n, j)binomial(k, j)j!.