A082545 -id:A082545 - cp's OEIS frontend

A343832 a(n) = Sum_{k=0..n} k! * binomial(n,k) * binomial(2*n+1,k).

1, 4, 31, 358, 5509, 106096, 2456299, 66471826, 2059640713, 71920704124, 2794938616471, 119653108240414, 5595650767265101, 283841520215780008, 15523069639558351459, 910529206043204428426, 57023540590242398853649, 3797750659849704886903156, 268025698704886063968108943

Offset: 0

Seiichi Manyama, May 01 2021

nonn

Let A(x) be the e.g.f. of this sequence, and B(x) be the e.g.f. of A082545, then A(x)/B(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..365
Wikipedia, Laguerre polynomials
Index entries for sequences related to Laguerre polynomials

Cf. A000522, A045721, A082545, A152059, A248668, A343830, A343831, A000108.

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

a := n -> add(k!*binomial(n, k)*binomial(2*n+1, k), k=0..n):
a := n -> n!*add(binomial(2*n+1, k)/(n-k)!, k=0..n):
a := n -> (-1)^n*KummerU(-n, n+2, -1):
a := n -> n!*LaguerreL(n, n+1, -1): # Peter Luschny, May 02 2021

a[n_] := Sum[k! * Binomial[n, k] * Binomial[2*n+1, k], {k, 0, n}]; Array[a, 20, 0] (* Amiram Eldar, May 01 2021 *)
Table[(-1)^n * HypergeometricU[-n, 2 + n, -1], {n, 0, 20}] (* Vaclav Kotesovec, May 02 2021 *)

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

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

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

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

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

a(n) = (2*n+1)! * Sum_{k=0..n} binomial(n,k)/(k+n+1)!.
a(n) = n! * Sum_{k=0..n} 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) == 1 (mod 3).
a(n) ~ 2^(2*n + 3/2) * n^n / exp(n-1). - Vaclav Kotesovec, May 02 2021
From Paul D. Hanna, Aug 16 2022: (Start)
E.g.f.: exp( (1-2*x - sqrt(1-4*x))/(2*x) ) * (1 - sqrt(1-4*x)) / (2*x*sqrt(1-4*x)), derived from the e.g.f for A082545 given by Mark van Hoeij.
E.g.f.: exp(C(x) - 1) * C(x) / sqrt(1-4*x), where C(x) = (1 - sqrt(1-4*x))/(2*x) is the Catalan function (A000108). (End)

A006902 a(n) = (2n)! * Sum_{k=0..n} (-1)^k * binomial(n,k) / (n+k)!.

Original entry on oeis.org

1, 1, 5, 47, 641, 11389, 248749, 6439075, 192621953, 6536413529, 248040482741, 10407123510871, 478360626529345, 23903857657114837, 1290205338991689821, 74803882225482661259, 4636427218380366565889, 305927317398343461908785, 21410426012751471702223333

Offset: 0

Simon Plouffe and N. J. A. Sloane

Number of words on {1,1,2,2,...,n,n} with longest increasing subsequence of length n. a(2) = 5: 1122, 1212, 1221, 2112, 2121. - Alois P. Heinz, Jan 18 2016

J. D. Horton and A. Kurn, Counting sequences with complete increasing subsequences, Congress. Numerantium, 33 (1981), 75-80.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..360 (terms n = 1..100 from T. D. Noe)
Yassine El Maazouz and Jim Pitman, The Bernoulli clock: probabilistic and combinatorial interpretations of the Bernoulli polynomials by circular convolution, arXiv:2210.02027 [math.PR], 2022.

Row n=2 of A047909.
Main diagonal of A267480.
Cf. A082545.

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

a:= proc(n) option remember; `if`(n<3, [1$2, 5][n+1], (
      (n^3+n^2-7*n+4)*a(n-1)-2*(2*n-3)*(n-1)^3*a(n-2))/(n-2))
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Jan 15 2016

Table[(-1)^k HypergeometricU[-k, 1+k, 1], {k,0,20}] (* Vladimir Reshetnikov, Feb 16 2011 *)

a(n)=round(hyperu(-n,n+1,1)*(-1)^n) \\ Charles R Greathouse IV, Dec 30 2014

[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). - Vladeta Jovovic, May 11 2003
(n-2)*a(n) - (n^3+n^2-7*n+4)*a(n-1) + 2*(2*n-3)*(n-1)^3*a(n-2) = 0. - Vladeta Jovovic, Jul 16 2004
a(n) ~ n^n*2^(2*n+1/2)/exp(n+1). - Vaclav Kotesovec, Jun 22 2013
a(n) = B_n(n*0!,(n-1)*1!, ..., 1*(n-1)!), where B_n(x1, ..., xn) is the n-th complete Bell polynomial. - Max Alekseyev, Jul 04 2015
a(n) = n!*binomial(2*n,n)*hypergeom([-n], [n+1], 1). - Peter Luschny, May 04 2017
a(n) = n!*Z(S_n; n, n-1, ..., 1) where Z(S_n) is the cycle index of the symmetric group of order n. - Sean A. Irvine, Nov 14 2017
a(n) = n! * [x^n] exp(-x/(1 - x))/(1 - x)^(n+1). - Ilya Gutkovskiy, Nov 21 2017
E.g.f.: exp(1-c(x))/sqrt(1-4*x), where c(x) = (1-sqrt(1-4*x))/(2*x) is the Catalan number generating function. - Ira M. Gessel, Jun 04 2021

a(0)=1 prepended by Alois P. Heinz, Jan 15 2016

A295385 a(n) = n!Sum_{k=0..n} binomial(2n,n-k)*n^k/k!.

Original entry on oeis.org

1, 3, 32, 579, 14736, 483115, 19376928, 918980139, 50306339072, 3121729082739, 216541483852800, 16603614676249843, 1394473165806440448, 127308860552307549531, 12553171419275174137856, 1329537514269062031406875, 150531055969843353812533248, 18143286205523964035258551651

Offset: 0

Ilya Gutkovskiy, Nov 21 2017

nonn

G. C. Greubel, Table of n, a(n) for n = 0..335
Eric Weisstein's World of Mathematics, Laguerre Polynomial
Wikipedia, Laguerre polynomials
Index entries for sequences related to Laguerre polynomials

Cf. A082545, A277373, A277423, A295384, A295406, A295407, A295408, A295409.

[Factorial(n)*(&+[Binomial(2*n,n-k)*n^k/Factorial(k): k in [0..n]]): n in [0..30]]; // G. C. Greubel, Feb 06 2018

Table[n! SeriesCoefficient[Exp[n x/(1 - x)]/(1 - x)^(n + 1), {x, 0, n}], {n, 0, 17}]
Table[n! LaguerreL[n, n, -n], {n, 0, 17}]
Table[(-1)^n HypergeometricU[-n, n + 1, -n], {n, 0, 17}]
Join[{1}, Table[n! Sum[Binomial[2 n, n - k] n^k/k!, {k, 0, n}], {n, 1, 17}]]

for(n=0,30, print1(n!*sum(k=0,n, binomial(2*n,n-k)*n^k/k!), ", ")) \\ G. C. Greubel, Feb 06 2018

a(n) = n! * [x^n] exp(n*x/(1 - x))/(1 - x)^(n+1).
a(n) = n!*Laguerre(n,n,-n).
a(n) ~ 2^(n - 1/2) * (1 + sqrt(2))^(n + 1/2) * n^n / exp((2 - sqrt(2))*n). - Vaclav Kotesovec, 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)

A343861 Coefficient triangle of generalized Laguerre polynomials n!*L(n,n,x) (rising powers of x).

Original entry on oeis.org

1, 2, -1, 12, -8, 1, 120, -90, 18, -1, 1680, -1344, 336, -32, 1, 30240, -25200, 7200, -900, 50, -1, 665280, -570240, 178200, -26400, 1980, -72, 1, 17297280, -15135120, 5045040, -840840, 76440, -3822, 98, -1, 518918400, -461260800, 161441280, -29352960, 3057600, -188160, 6720, -128, 1

Offset: 0

Seiichi Manyama, May 01 2021

			The triangle begins:
       1;
       2,      -1;
      12,      -8,      1;
     120,     -90,     18,     -1;
    1680,   -1344,    336,    -32,    1;
   30240,  -25200,   7200,   -900,   50,  -1;
  665280, -570240, 178200, -26400, 1980, -72, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Wikipedia, Laguerre polynomials
Index entries for sequences related to Laguerre polynomials

For k=0..1 the (unsigned) columns give A001813, A092956(n-1).
Row sums (signed) give A006902, row sums (unsigned) give A082545.
Cf. A066667, A062137, A062138, A062139, A062140.

[(-1)^k*Factorial(n-k)*Binomial(n,k)*Binomial(2*n, n+k): k in [0..n], n in [0..12]]; // G. C. Greubel, Aug 11 2022

T[n_, k_] := (-1)^k * (2*n)! * Binomial[n, k]/(k + n)!; Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Amiram Eldar, May 11 2021 *)

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

PARI
```
row(n) = Vecrev(n!*pollaguerre(n, n));
```

def A343861(n,k): return (-1)^k*factorial(n-k)*binomial(n,k)*binomial(2*n,n+k)
flatten([[A343861(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Aug 11 2022

T(n, k) = (-1)^k * n! * binomial(2*n,n-k)/k! = (-1)^k * (2*n)! * binomial(n,k)/(k+n)!.
T(n, 0) = A001813(n).
T(n, 1) = -A092956(n-1).
Sum_{k=0..n} T(n, k) = A006902(n).
Sum_{k=0..n} (-1)^k * T(n, k) = A082545(n).

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).

A253670 Square array read by ascending antidiagonals, T(n, k) = k![x^k](exp(x)sum(j=0..n, C(2n,j)x^j)), n>=0, k>=0.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 5, 5, 1, 1, 7, 21, 7, 1, 1, 9, 43, 49, 9, 1, 1, 11, 73, 229, 89, 11, 1, 1, 13, 111, 529, 685, 141, 13, 1, 1, 15, 157, 1021, 3393, 1531, 205, 15, 1, 1, 17, 211, 1753, 8501, 12361, 2887, 281, 17, 1, 1, 19, 273, 2773, 18001, 63591, 32809, 4873, 369, 19, 1

Offset: 0

Peter Luschny, Jan 18 2015

			Square array starts:
[n\k][0   1    2     3      4       5        6]
[0]   1,  1,   1,    1,     1,      1,       1, ...
[1]   1,  3,   5,    7,     9,     11,      13, ...
[2]   1,  5,  21,   49,    89,    141,     205, ...
[3]   1,  7,  43,  229,   685,   1531,    2887, ...
[4]   1,  9,  73,  529,  3393,  12361,   32809, ...
[5]   1, 11, 111, 1021,  8501,  63591,  272851, ...
[6]   1, 13, 157, 1753, 18001, 169021, 1442173, ...
The first few rows as a triangle:
1
1,  1
1,  3,  1
1,  5,  5,   1
1,  7, 21,   7,  1
1,  9, 43,  49,  9,  1
1, 11, 73, 229, 89, 11, 1

Cf. A082545.

T := (n,k) -> k!*coeff(series(exp(x)*add(binomial(2*n,j)*x^j,j=0..n), x, k+1), x, k): for n from 0 to 6 do lprint(seq(T(n,k), k=0..6)) od;

T(n,n) = A082545(n).

cp's OEIS Frontend

A343832 a(n) = Sum_{k=0..n} k! * binomial(n,k) * binomial(2*n+1,k).

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A006902 a(n) = (2n)! * Sum_{k=0..n} (-1)^k * binomial(n,k) / (n+k)!.

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A295385 a(n) = n!*Sum_{k=0..n} binomial(2*n,n-k)*n^k/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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A343861 Coefficient triangle of generalized Laguerre polynomials n!*L(n,n,x) (rising powers of x).

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A295385 a(n) = n!Sum_{k=0..n} binomial(2n,n-k)*n^k/k!.

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

A253670 Square array read by ascending antidiagonals, T(n, k) = k![x^k](exp(x)sum(j=0..n, C(2n,j)x^j)), n>=0, k>=0.