A277373 -id:A277373 - cp's OEIS frontend

A340863 a(n) = n!*LaguerreL(n, -n^2).

1, 2, 34, 1626, 151064, 23046370, 5228520912, 1651548277946, 692979602529664, 372856154213080674, 250277853396112428800, 205025892171407329263802, 201314381459222197472984064, 233396220344077025321595074306

Offset: 0

Seiichi Manyama, Feb 05 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..214
Eric Weisstein's World of Mathematics, Laguerre Polynomial
Wikipedia, Laguerre polynomials
Index entries for sequences related to Laguerre polynomials

Main diagonal of A338435.

Cf. A277373, A289192.

Table[n! * LaguerreL[n, -n^2], {n, 0, 13}] (* Amiram Eldar, Feb 05 2021 *)

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

a(n) = n!*pollaguerre(n, 0, -n^2); \\ Michel Marcus, Feb 05 2021

a(n) = Sum_{k=0..n} n^(2*k) * (n-k)! * binomial(n,k)^2.
a(n) = n! * [x^n] exp(n^2 * x/(1-x))/(1-x).
a(n) = A289192(n,n^2).
a(n) ~ exp(1) * n^(2*n). - Vaclav Kotesovec, Feb 14 2021

A341197 a(n) = Sum_{k=0..n} k^n * (n-k)! * binomial(n,k)^2.

Original entry on oeis.org

1, 1, 8, 117, 2800, 97125, 4551876, 274975897, 20690260928, 1889451727497, 205192914235300, 26068434774065541, 3822244304373085680, 639508508456359098349, 120922358829574588363364, 25626415609908102483018225

Offset: 0

Seiichi Manyama, Feb 06 2021

nonn

Main diagonal of A341200.

Cf. A277373, A341185.

a[0] = 1; a[n_] := Sum[k^n * (n-k)! * Binomial[n, k]^2, {k, 0, n}]; Array[a, 16, 0] (* Amiram Eldar, Feb 06 2021 *)

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

A343847 T(n, k) = (n - k)! * [x^(n-k)] exp(k*x/(1 - x))/(1 - x). Triangle read by rows, T(n, k) for 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 6, 7, 3, 1, 24, 34, 14, 4, 1, 120, 209, 86, 23, 5, 1, 720, 1546, 648, 168, 34, 6, 1, 5040, 13327, 5752, 1473, 286, 47, 7, 1, 40320, 130922, 58576, 14988, 2840, 446, 62, 8, 1, 362880, 1441729, 671568, 173007, 32344, 4929, 654, 79, 9, 1

Offset: 0

Peter Luschny, May 07 2021

			Triangle starts:
0:     1;
1:     1,      1;
2:     2,      2,     1;
3:     6,      7,     3,     1;
4:    24,     34,    14,     4,    1;
5:   120,    209,    86,    23,    5,   1;
6:   720,   1546,   648,   168,   34,   6,  1;
7:  5040,  13327,  5752,  1473,  286,  47,  7,  1;
8: 40320, 130922, 58576, 14988, 2840, 446, 62,  8,  1;
.
Array whose upward read antidiagonals are the rows of the triangle.
n\k   0       1       2        3        4         5        6
-----------------------------------------------------------------
0:    1,      1,      1,       1,       1,        1,        1, ...
1:    1,      2,      3,       4,       5,        6,        7, ...
2:    2,      7,     14,      23,      34,       47,       62, ...
3:    6,     34,     86,     168,     286,      446,      654, ...
4:   24,    209,    648,    1473,    2840,     4929,     7944, ...
5:  120,   1546,   5752,   14988,   32344,    61870,   108696, ...
6:  720,  13327,  58576,  173007,  414160,   866695,  1649232, ...
7: 5040, 130922, 671568, 2228544, 5876336, 13373190, 27422352, ...

Row sums: A343848. T(2*n, n) = A277373(n). Variant: A289192.
Columns: A000142, A002720, A087912, A277382, A289147, A289211, A289212, A289213, A289214, A289215, A289216.
Cf. A021009 (Laguerre polynomials), A344048.

T := proc(n, k) option remember;
if n = k then return 1 elif n = k+1 then return k+1 fi;
(2*n-k-1)*T(n-1, k) - (n-k-1)^2*T(n-2, k) end:
seq(print(seq(T(n ,k), k = 0..n)), n = 0..7);

T[n_, k_] := (-1)^(n - k) HypergeometricU[k - n, 1, -k];
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten
(* Alternative: *)
TL[n_, k_] := (n - k)! LaguerreL[n - k, -k];
Table[TL[n, k], {n, 0, 9}, {k, 0, n}] // Flatten

T(n, k) = (n - k)!*sum(j=0, n - k, binomial(n - k, j) * k^j / j!)
for(n=0, 9, for(k=0, n, print(T(n, k))))

# Columns of the array.
def column(k, len):
    R. = PowerSeriesRing(QQ, default_prec=len)
    f = exp(k * x / (1 - x)) / (1 - x)
    return f.egf_to_ogf().list()
for col in (0..6): print(column(col, 20))

T(n, k) = (-1)^(n - k)*U(k - n, 1, -k), where U is the Kummer U function.
T(n, k) = (n - k)! * L(n - k, -k), where L is the Laguerre polynomial function.
T(n, k) = (n - k)! * Sum_{j = 0..n - k} binomial(n - k, j) k^j / j!.
T(n, k) = (2*n-k-1)*T(n-1, k) - (n-k-1)^2*T(n-2, k) for n - k >= 2.

A344106 a(n) = n! * LaguerreL(n, -n+1).

Original entry on oeis.org

1, 1, 7, 86, 1473, 32344, 866695, 27422352, 1000578817, 41361536384, 1910451937671, 97512721964800, 5450486787062977, 331112639931669504, 21722219855305516807, 1530517712811373819904, 115269154497700063898625, 9241045907270523509112832, 785719407951447904088069383

Offset: 0

Vaclav Kotesovec, May 09 2021

nonn

Cf. A001622, A277373, A343849, A344107.

Table[n!*LaguerreL[n, -n+1], {n, 0, 20}]

a(n) = n!*subst(pollaguerre(n), x, 1-n); \\ Michel Marcus, May 09 2021

a(n) ~ exp((n-1)/phi - n) * phi^(2*n+1) * n^n / 5^(1/4), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio.

a(n) = Sum_{k=0..n} binomial(n, k)^2*(n - k)!*(n - 1)^k. - Peter Luschny, Dec 25 2021

A344107 a(n) = n! * LaguerreL(n, -n+2).

Original entry on oeis.org

1, 0, 2, 34, 648, 14988, 414160, 13373190, 495057024, 20686611736, 963532779264, 49510386761130, 2782552712473600, 169808382346687524, 11182975029610555392, 790535958617173397902, 59708339321680507207680, 4798728602043471360126000, 408910082803875174679773184

Offset: 0

Vaclav Kotesovec, May 09 2021

nonn

In general, for fixed k>=0, n!*LaguerreL(n, -n+k) ~ exp((n-k)/phi - n) * phi^(2*n+1) * n^n / 5^(1/4).

Cf. A001622, A277373, A343849, A344106.

Table[n!*LaguerreL[n, -n+2], {n, 0, 20}]

a(n) = n!*subst(pollaguerre(n), x, 2-n); \\ Michel Marcus, May 09 2021

a(n) ~ exp((n-2)/phi - n) * phi^(2*n+1) * n^n / 5^(1/4), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio.

A277453 a(n) = Sum_{k=0..n} binomial(n,k) * 2^k * n^k * k!.

Original entry on oeis.org

1, 3, 41, 1531, 111393, 13262051, 2336744233, 570621092091, 184341785557121, 76092709735150723, 39064090158380196201, 24408768326642565035963, 18237590837527919131499041, 16056004231253610384348995811, 16448689708899063469247204152553

Offset: 0

Vaclav Kotesovec, Oct 16 2016

nonn

Cf. A086331, A277373, A277423, A277452.

Flatten[{1, Table[Sum[Binomial[n, k]*2^k*n^k*k!, {k, 0, n}], {n, 1, 20}]}]

a(n) = exp(1/(2*n)) * 2^n * n^n * Gamma(n+1, 1/(2*n)).

a(n) ~ 2^n * n^n * n!.

A338435 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = n!LaguerreL(n, -kn).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 14, 6, 1, 4, 34, 168, 24, 1, 5, 62, 654, 2840, 120, 1, 6, 98, 1626, 17688, 61870, 720, 1, 7, 142, 3246, 59928, 616120, 1649232, 5040, 1, 8, 194, 5676, 151064, 2844120, 26252496, 51988748, 40320, 1, 9, 254, 9078, 318744, 9052120, 165100752, 1322624016, 1891712384, 362880

Offset: 0

Seiichi Manyama, Feb 05 2021

			Square array begins:
   1,    1,     1,     1,      1, ...
   1,    2,     3,     4,      5, ...
   2,   14,    34,    62,     98, ...
   6,  168,   654,  1626,   3246, ...
  24, 2840, 17688, 59928, 151064, ...

Eric Weisstein's World of Mathematics, Laguerre Polynomial
Wikipedia, Laguerre polynomials
Index entries for sequences related to Laguerre polynomials

Columns k=0..8 gives A000142, A277373, A277391, A277392, A277418, A277419, A277420, A277421, A277422.
Main diagonal gives A340863.
Cf. A021009, A289192 (n!*LaguerreL(n, -k)), A341014.

T[n_, k_] := n! * LaguerreL[n, -k*n]; Table[T[k, n - k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, Feb 05 2021 *)

T(n, k) = sum(j=0, n, (k*n)^j*(n-j)!*binomial(n, j)^2);

T(n, k) = n!*pollaguerre(n, 0, -k*n); \\ Michel Marcus, Feb 05 2021

T(n,k) = Sum_{j=0..n} (k*n)^j * (n-j)! * binomial(n,j)^2.
T(n,k) = n! * [x^n] exp(k*n*x/(1-x))/(1-x).
T(n,k) = A289192(n,k*n).

A354943 a(n) = Sum_{k=0..n} binomial(n,k)^3 * k! * n^(n-k).

Original entry on oeis.org

1, 2, 22, 438, 12824, 496370, 23914512, 1379269094, 92667551104, 7108231236066, 612974464428800, 58702772664490262, 6181602019316333568, 709911177607125141362, 88301595129435811723264, 11825985945777638231211750, 1696696168760520436580974592, 259624546758869333450285984066

Offset: 0

Ilya Gutkovskiy, Jun 12 2022

nonn

Cf. A000172, A063170, A216831, A241247, A274246, A277373, A277386, A354944.

Unprotect[Power]; 0^0 = 1; Table[Sum[Binomial[n, k]^3 k! n^(n - k), {k, 0, n}], {n, 0, 17}]
Unprotect[Power]; 0^0 = 1; Table[n!^3 SeriesCoefficient[BesselI[0, 2 Sqrt[x]] Sum[n^k x^k/k!^3, {k, 0, n}], {x, 0, n}], {n, 0, 17}]

a(n) = sum(k=0, n, binomial(n,k)^3 * k! * n^(n-k)); \\ Michel Marcus, Jun 12 2022

a(n) = n!^3 * [x^n] BesselI(0,2*sqrt(x)) * Sum_{k>=0} n^k * x^k / k!^3.

a(n) ~ c * n^(n - 1/2) / (exp(r*n) * r^(2*n)), where r = (2 - 5*(2/(3*sqrt(69)-11))^(1/3) + ((3*sqrt(69)-11)/2)^(1/3))/3 = 0.430159709001946734... is the real root of the equation r^2 = (1-r)^3 and c = sqrt(138 + 2^(2/3)*(69*(8901 - 223*sqrt(69)))^(1/3) + 2^(2/3)*(69*(8901 + 223*sqrt(69)))^(1/3))/(2*sqrt(69*Pi)) = 0.684738330749970434111338151096549475398274404060139170789278633219363118... - Vaclav Kotesovec, Jul 01 2022, updated Mar 17 2024

A341056 a(n) = n! * [x^n] exp(x/(1 - n*x)) / (1 - x).

Original entry on oeis.org

1, 2, 9, 106, 2801, 132426, 9705577, 1015001954, 143392421601, 26298332570386, 6074043257989001, 1724846814877790682, 590605908915568818769, 239956225437223244619866, 114123836188192016600789481, 62808518765936960824453590226, 39603421893790601518269204039617

Offset: 0

Seiichi Manyama, Feb 04 2021

nonn

			a(3) = 3! * (1 + 1/1! + 7/2! + 73/3!) = 106.

Cf. A277373, A293146, A330260, A331658, A341033.

Table[n!*(1 + Sum[Sum[n^(j-k)*Binomial[j-1, k-1]/k!, {k, 1, j}], {j, 1, n}]), {n, 0, 20}] (* Vaclav Kotesovec, Feb 14 2021 *)

{a(n) = n!*(1+sum(j=1, n, sum(k=1, j, n^(j-k)*binomial(j-1, k-1)/k!)))}

a(n) = n! * Sum_{k=0..n} A341033(k,n)/k! = n! * (1 + Sum_{j=1.. n} Sum_{k=1.. j} n^(j-k) * binomial(j-1,k-1)/k!).

a(n) ~ BesselI(1,2) * n! * n^(n-1). - Vaclav Kotesovec, Feb 14 2021

cp's OEIS Frontend

A340863 a(n) = n!*LaguerreL(n, -n^2).

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A341197 a(n) = Sum_{k=0..n} k^n * (n-k)! * binomial(n,k)^2.

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A343847 T(n, k) = (n - k)! * [x^(n-k)] exp(k*x/(1 - x))/(1 - x). Triangle read by rows, T(n, k) for 0 <= k <= n.

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A344106 a(n) = n! * LaguerreL(n, -n+1).

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A344107 a(n) = n! * LaguerreL(n, -n+2).

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A277453 a(n) = Sum_{k=0..n} binomial(n,k) * 2^k * n^k * k!.

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A338435 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = n!*LaguerreL(n, -k*n).

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A354943 a(n) = Sum_{k=0..n} binomial(n,k)^3 * k! * n^(n-k).

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A338435 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = n!LaguerreL(n, -kn).