cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 11-16 of 16 results.

A289213 a(n) = n! * Laguerre(n,-7).

Original entry on oeis.org

1, 8, 79, 916, 12113, 179152, 2921911, 51988748, 1000578817, 20686611736, 456805020959, 10721879413252, 266382974861521, 6980304560060384, 192311632290456007, 5555079068684580988, 167822887344661475969, 5290815252203206305832, 173713426149927498289903
Offset: 0

Views

Author

Alois P. Heinz, Jun 28 2017

Keywords

Links

Crossrefs

Column k=7 of A289192.
Cf. A160607, A160608.

Programs

  • Magma
    m:=25; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(7*x/(1-x))/(1-x))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, May 13 2018
  • Maple
    a:= n-> n! * add(binomial(n, i)*7^i/i!, i=0..n):
seq(a(n), n=0..20);
  • Mathematica
    Table[n!*LaguerreL[n, -7], {n, 0, 20}] (* Indranil Ghosh, Jul 04 2017 *)
  • PARI
    x = 'x + O('x^30); Vec(serlaplace(exp(7*x/(1-x))/(1-x))) \\ Michel Marcus, Jul 04 2017
  • Python
    from mpmath import *
mp.dps=100
def a(n): return int(fac(n)*laguerre(n, 0, -7))
print([a(n) for n in range(21)]) # Indranil Ghosh, Jul 04 2017

Formula

E.g.f.: exp(7*x/(1-x))/(1-x).
a(n) = n! * Sum_{i=0..n} 7^i/i! * binomial(n,i).
a(n) = n! * A160607(n)/A160608(n).
a(n) ~ exp(-7/2 + 2*sqrt(7*n) - n) * n^(n + 1/4) / (sqrt(2)*7^(1/4)) * (1 + 367/(48*sqrt(7*n))). - Vaclav Kotesovec, Nov 13 2017
Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(x) * Sum_{n>=0} 7^n * x^n / (n!)^2. - Ilya Gutkovskiy, Jul 17 2020

A295381 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(-k*x/(1 - x))/(1 - x).

Original entry on oeis.org

1, 1, 1, 1, 0, 2, 1, -1, -1, 6, 1, -2, -2, -4, 24, 1, -3, -1, -2, -15, 120, 1, -4, 2, 6, 8, -56, 720, 1, -5, 7, 14, 33, 88, -185, 5040, 1, -6, 14, 16, 24, 102, 592, -204, 40320, 1, -7, 23, 6, -31, -104, -9, 3344, 6209, 362880, 1, -8, 34, -22, -120, -380, -1328, -3762, 14464, 112400, 3628800
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 21 2017

Keywords

Examples

			E.g.f. of column k: A_k(x) = 1 + (1 - k)*x/1! + (k^2 - 4*k + 2)*x^2/2! + (-k^3 + 9*k^2 - 18*k + 6)*x^3/3! + (k^4 - 16*k^3 + 72*k^2 - 96*k + 24)*x^4/4! + ...
Square array begins:
    1,   1,   1,    1,    1,    1, ...
    1,   0,  -1,   -2    -3,   -4, ...
    2,  -1,  -2,   -1,    2,    7, ...
    6,  -4,  -2,    6,   14,   16, ...
   24, -15,   8,   33,   24,  -31, ...
  120, -56,  88,  102, -104, -380, ...

Links

Crossrefs

Columns k=0..2 give A000142, A009940, A295382.
Main diagonal gives A277423.
Cf. A289192.

Programs

  • Mathematica
    Table[Function[k, n! SeriesCoefficient[Exp[-k x/(1 - x)]/(1 - x), {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten
Table[Function[k, n! LaguerreL[n, k]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten
Table[Function[k, n! Hypergeometric1F1[-n, 1, k]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten

Formula

E.g.f. of column k: exp(-k*x/(1 - x))/(1 - x).
A(n,k) = n!*Laguerre(n,k).

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

Original entry on oeis.org

1, 2, 34, 1626, 151064, 23046370, 5228520912, 1651548277946, 692979602529664, 372856154213080674, 250277853396112428800, 205025892171407329263802, 201314381459222197472984064, 233396220344077025321595074306
Offset: 0

Views

Author

Seiichi Manyama, Feb 05 2021

Keywords

Links

Crossrefs

Main diagonal of A338435.
Cf. A277373, A289192.

Programs

  • Mathematica
    Table[n! * LaguerreL[n, -n^2], {n, 0, 13}] (* Amiram Eldar, Feb 05 2021 *)
  • PARI
    a(n) = sum(k=0, n, n^(2*k)*(n-k)!*binomial(n, k)^2);
  • PARI
    a(n) = n!*pollaguerre(n, 0, -n^2); \\ Michel Marcus, Feb 05 2021

Formula

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

A341200 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = Sum_{j=0..n} j^k * (n-j)! * binomial(n,j)^2.

Original entry on oeis.org

1, 0, 2, 0, 1, 7, 0, 1, 6, 34, 0, 1, 8, 39, 209, 0, 1, 12, 63, 292, 1546, 0, 1, 20, 117, 544, 2505, 13327, 0, 1, 36, 243, 1168, 5225, 24306, 130922, 0, 1, 68, 549, 2800, 12525, 55656, 263431, 1441729, 0, 1, 132, 1323, 7312, 33425, 145836, 653023, 3154824, 17572114
Offset: 0

Views

Author

Seiichi Manyama, Feb 06 2021

Keywords

Examples

			Square array begins:
     1,    0,    0,     0,     0,     0, ...
     2,    1,    1,     1,     1,     1, ...
     7,    6,    8,    12,    20,    36, ...
    34,   39,   63,   117,   243,   549, ...
   209,  292,  544,  1168,  2800,  7312, ...
  1546, 2505, 5225, 12525, 33425, 97125, ...

Crossrefs

Columns k=0..4 gives A002720, A103194, A105219, A105218, A341196.
Main diagonal gives A341197.
Cf. A289192.

Programs

  • Mathematica
    T[n_, k_] := Sum[If[j == k == 0, 1, j^k] * (n - j)! * Binomial[n, j]^2, {j, 0, n}]; Table[T[k, n - k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, Feb 06 2021 *)
  • PARI
    T(n, k) = sum(j=0, n, j^k*(n-j)!*binomial(n, j)^2);

Formula

About e.g.f. of column k, see A105218 or A105219 comment.

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

Views

Author

Peter Luschny, May 07 2021

Keywords

Examples

			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, ...

Crossrefs

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.

Programs

  • Maple
    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);
  • Mathematica
    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
  • PARI
    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))))
  • SageMath
    # 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))

Formula

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.

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

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

Views

Author

Seiichi Manyama, Feb 05 2021

Keywords

Examples

			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, ...

Links

Crossrefs

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.

Programs

  • Mathematica
    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 *)
  • PARI
    T(n, k) = sum(j=0, n, (k*n)^j*(n-j)!*binomial(n, j)^2);
  • PARI
    T(n, k) = n!*pollaguerre(n, 0, -k*n); \\ Michel Marcus, Feb 05 2021

Formula

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).
Previous Showing 11-16 of 16 results.

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