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.

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A301419 a(n) = [x^n] Sum_{k>=0} x^k/Product_{j=1..k} (1 - n*j*x).

Original entry on oeis.org

1, 1, 3, 19, 201, 3176, 69823, 2026249, 74565473, 3376695763, 183991725451, 11854772145800, 890415496931689, 77023751991841669, 7592990698770559111, 845240026276785888451, 105409073489605774592897, 14625467507717709778793020, 2244123413703647502288608467, 378751257186051653931253015229
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 20 2018

Keywords

Links

Crossrefs

Cf. A000110, A004211, A004212, A004213, A005011, A005012, A008277, A075506, A075507, A075508, A075509, A242817, A292914, A318183.

Programs

  • GAP
    List([0..20],n->Sum([0..n],k->n^(n-k)*Stirling2(n,k))); # Muniru A Asiru, Mar 20 2018
  • Mathematica
    Table[SeriesCoefficient[Sum[x^k/Product[(1 - n j x), {j, 1, k}], {k, 0, n}], {x, 0, n}], {n, 0, 19}]
Join[{1}, Table[n! SeriesCoefficient[Exp[(Exp[n x] - 1)/n], {x, 0, n}], {n, 19}]]
Join[{1}, Table[Sum[n^(n - k) StirlingS2[n, k], {k, 0, n}], {n, 19}]]
(* Or: *)
A301419[n_] := If[n == 0, 1, n^n BellB[n, 1/n]];
Table[A301419[n], {n, 0, 19}] (* Peter Luschny, Dec 22 2021 *)
  • PARI
    a(n) = sum(k=0, n, n^(n-k)*stirling(n, k, 2)); \\ Michel Marcus, Mar 23 2018

Formula

a(n) = n! * [x^n] exp((exp(n*x) - 1)/n), for n > 0.
a(n) = Sum_{k=0..n} n^(n-k)*Stirling2(n,k).
a(n) = n^n * BellPolynomial(n, 1/n) for n >= 1. - Peter Luschny, Dec 22 2021
a(n) ~ exp(n/LambertW(n^2) - n) * n^(2*n) / (sqrt(1 + LambertW(n^2)) * LambertW(n^2)^n). - Vaclav Kotesovec, Jun 06 2022

A049402 Row sums of triangle A049374.

Original entry on oeis.org

1, 1, 7, 61, 649, 8245, 122215, 2069425, 39328465, 827226505, 19047582055, 475956135205, 12815133759385, 369605936607805, 11361372997850695, 370609338222772825, 12780705695068446625, 464412124831585889425, 17729002673226394402375, 709180766131239680070925
Offset: 0

Views

Author

Wolfdieter Lang

Keywords

Links

Crossrefs

Column k=6 of A291709.
Cf. A049374, A049427.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 1, add(
      binomial(n-1, j-1)*(j+4)!/5!*a(n-j), j=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Aug 01 2017
  • Mathematica
    a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, j - 1]*(j + 4)!/5!*a[n - j], {j, 1, n}];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Jun 04 2018, after Alois P. Heinz *)

Formula

E.g.f. exp(p(x)) with p(x) := (1-(1-x)^5)/(5*(1-x)^5) (E.g.f. first column of A049374).
From Seiichi Manyama, Jan 18 2025: (Start)
a(n) = Sum_{k=0..n} |Stirling1(n,k)| * A005011(k).
a(n) = (1/exp(1/5)) * (-1)^n * n! * Sum_{k>=0} binomial(-5*k,n)/(5^k * k!). (End)

Extensions

a(0)=1 prepended by Alois P. Heinz, Aug 01 2017

A241578 Square array read by antidiagonals upwards: T(n,k) = Sum_{j=1..k} n^(k-j)*Stirling_2(k,j) (n >= 0, k >= 1).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 5, 1, 1, 4, 11, 15, 1, 1, 5, 19, 49, 52, 1, 1, 6, 29, 109, 257, 203, 1, 1, 7, 41, 201, 742, 1539, 877, 1, 1, 8, 55, 331, 1657, 5815, 10299, 4140, 1, 1, 9, 71, 505, 3176, 15821, 51193, 75905, 21147, 1, 1, 10, 89, 729, 5497, 35451, 170389, 498118, 609441, 115975, 1
Offset: 0

Views

Author

N. J. A. Sloane, Apr 29 2014

Keywords

Examples

			Array begins:
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115975, 678570, 4213597, ...
1, 3, 11, 49, 257, 1539, 10299, 75905, 609441, 5284451, 49134923, 487026929, ...
1, 4, 19, 109, 742, 5815, 51193, 498118, 5296321, 60987817, 754940848, 9983845261, ...
1, 5, 29, 201, 1657, 15821, 170389, 2032785, 26546673, 376085653, 5736591885, 93614616409, ...
1, 6, 41, 331, 3176, 35451, 447981, 6282416, 96546231, 1611270851, 28985293526, 558413253581, ...
1, 7, 55, 505, 5497, 69823, 1007407, 16157905, 284214097, 5432922775, 112034017735, 2476196276617, ...
1, 8, 71, 729, 8842, 125399, 2026249, 36458010, 719866701, 15453821461, 358100141148, 8899677678109, ...
...

Links

Crossrefs

Three versions of this array are A111673, A241578, A241579.
Rows and columns give A000110, A004211, A004212, A004213, A005011, A005012, A028387, A241577.

Programs

  • Maple
    with(combinat):
T:=(n,k)->add(n^(k-j)*stirling2(k,j),j=1..k);
r:=n->[seq(T(n,k),k=1..12)];
for n from 0 to 8 do lprint(r(n)); od:

A337041 a(n) = exp(-1/5) * Sum_{k>=0} (5*k - 1)^n / (5^k * k!).

Original entry on oeis.org

1, 0, 5, 25, 200, 1875, 20625, 256250, 3534375, 53515625, 881468750, 15667578125, 298478828125, 6060493750000, 130542772265625, 2971013486328125, 71193375156250000, 1790666151318359375, 47145509926611328125, 1296156682961425781250, 37129279010879638671875
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 12 2020

Keywords

Crossrefs

Cf. A000296, A003577, A005011, A337038, A337039, A337040, A337042, A337043.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[Exp[(Exp[5 x] - 1)/5 - x], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k] 5^k a[n - k - 1], {k, 1, n - 1}]; Table[a[n], {n, 0, 20}]
Table[Sum[(-1)^(n - k) Binomial[n, k] 5^k BellB[k, 1/5], {k, 0, n}], {n, 0, 20}]

Formula

G.f. A(x) satisfies: A(x) = (1 - 5*x + x*A(x/(1 - 5*x))) / (1 - 4*x - 5*x^2).
G.f.: (1/(1 + x)) * Sum_{k>=0} (x/(1 + x))^k / Product_{j=1..k} (1 - 5*j*x/(1 + x)).
E.g.f.: exp((exp(5*x) - 1) / 5 - x).
a(0) = 1; a(n) = Sum_{k=1..n-1} binomial(n-1,k) * 5^k * a(n-k-1).
a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * A005011(k).
a(n) ~ 5^(n - 1/5) * n^(n - 1/5) * exp(n/LambertW(5*n) - n - 1/5) / (sqrt(1 + LambertW(5*n)) * LambertW(5*n)^(n - 1/5)). - Vaclav Kotesovec, Jun 26 2022

A337595 a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} binomial(n,k)^2 * k * 5^(k-1) * a(n-k).

Original entry on oeis.org

1, 1, 7, 76, 1359, 35620, 1256470, 57247765, 3259660095, 225795951580, 18644190437550, 1805220546542425, 202173130530484350, 25889773647793362425, 3754040522961719322325, 611181508958872398483625, 110903705593861290502897375, 22285223101687304853202923500, 4930523789420612133816212731750
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 02 2020

Keywords

Crossrefs

Cf. A005011, A337592, A337593, A337594, A337597.

Programs

  • Mathematica
    a[0] = 1; a[n_] := a[n] = (1/n) Sum[Binomial[n, k]^2 k 5^(k - 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]
nmax = 18; CoefficientList[Series[Exp[(BesselI[0, 2 Sqrt[5 x]] - 1)/5], {x, 0, nmax}], x] Range[0, nmax]!^2

Formula

Sum_{n>=0} a(n) * x^n / (n!)^2 = exp((BesselI(0,2*sqrt(5*x)) - 1) / 5).
Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(Sum_{n>=1} 5^(n-1) * x^n / (n!)^2).

A351151 G.f. A(x) satisfies: A(x) = 1 + x^2 * A(x/(1 - 5*x)) / (1 - 5*x).

Original entry on oeis.org

1, 0, 1, 5, 26, 145, 901, 6420, 52501, 480955, 4795626, 51066375, 576182001, 6879462680, 86955722401, 1162559359745, 16392133866026, 242734091500445, 3758825675820501, 60660434188558780, 1017770666417312501, 17725289455315892375, 320047193447632729626
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 02 2022

Keywords

Comments

Shifts 2 places left under 5th-order binomial transform.

Crossrefs

Cf. A000994, A005011, A351056, A351143, A351144, A351150, A351152.

Programs

  • Mathematica
    nmax = 22; A[] = 0; Do[A[x] = 1 + x^2 A[x/(1 - 5 x)]/(1 - 5 x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[1] = 0; a[n_] := a[n] = Sum[Binomial[n - 2, k] 5^k a[n - k - 2], {k, 0, n - 2}]; Table[a[n], {n, 0, 22}]

Formula

a(0) = 1, a(1) = 0; a(n) = Sum_{k=0..n-2} binomial(n-2,k) * 5^k * a(n-k-2).

A241579 Square array read by antidiagonals downwards: T(n,k) = Sum_{j=1..k} n^(k-j)*Stirling_2(k,j) (n >= 0, k >= 1).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 5, 3, 1, 1, 15, 11, 4, 1, 1, 52, 49, 19, 5, 1, 1, 203, 257, 109, 29, 6, 1, 1, 877, 1539, 742, 201, 41, 7, 1, 1, 4140, 10299, 5815, 1657, 331, 55, 8, 1, 1, 21147, 75905, 51193, 15821, 3176, 505, 71, 9, 1, 1, 115975, 609441, 498118, 170389, 35451, 5497, 729, 89, 10, 1
Offset: 0

Views

Author

N. J. A. Sloane, Apr 29 2014

Keywords

Examples

			Array begins:
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115975, 678570, 4213597, ...
1, 3, 11, 49, 257, 1539, 10299, 75905, 609441, 5284451, 49134923, 487026929, ...
1, 4, 19, 109, 742, 5815, 51193, 498118, 5296321, 60987817, 754940848, 9983845261, ...
1, 5, 29, 201, 1657, 15821, 170389, 2032785, 26546673, 376085653, 5736591885, 93614616409, ...
1, 6, 41, 331, 3176, 35451, 447981, 6282416, 96546231, 1611270851, 28985293526, 558413253581, ...
1, 7, 55, 505, 5497, 69823, 1007407, 16157905, 284214097, 5432922775, 112034017735, 2476196276617, ...
1, 8, 71, 729, 8842, 125399, 2026249, 36458010, 719866701, 15453821461, 358100141148, 8899677678109, ...
...

Links

Crossrefs

Three versions of this array are A111673, A241578, A241579.
Rows and columns give A000110, A004211, A004212, A004213, A005011, A005012, A028387, A241577.

Programs

  • Maple
    with(combinat):
T:=(n,k)->add(n^(k-j)*stirling2(k,j),j=1..k);
r:=n->[seq(T(n,k),k=1..12)];
for n from 0 to 8 do lprint(r(n)); od:

A351056 G.f. A(x) satisfies: A(x) = 1 + x + x^2 * A(x/(1 - 5*x)) / (1 - 5*x).

Original entry on oeis.org

1, 1, 1, 6, 36, 221, 1431, 10121, 80311, 718106, 7111976, 76201501, 868288401, 10438492181, 132166853861, 1763179150946, 24776241643056, 365971430085021, 5662954240306111, 91450179009971181, 1536249848608545451, 26782376261726525126, 483792982362049317676
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 30 2022

Keywords

Comments

Shifts 2 places left under 5th-order binomial transform.

Crossrefs

Cf. A005011, A007472, A007476, A351049, A351050, A351057.

Programs

  • Mathematica
    nmax = 22; A[] = 0; Do[A[x] = 1 + x + x^2 A[x/(1 - 5 x)]/(1 - 5 x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = a[1] = 1; a[n_] := a[n] = Sum[Binomial[n - 2, k] 5^k a[n - k - 2], {k, 0, n - 2}]; Table[a[n], {n, 0, 22}]

Formula

a(0) = a(1) = 1; a(n) = Sum_{k=0..n-2} binomial(n-2,k) * 5^k * a(n-k-2).

A351132 G.f. A(x) satisfies: A(x) = x + x^2 * A(x/(1 - 5*x)) / (1 - 5*x).

Original entry on oeis.org

0, 1, 0, 1, 10, 76, 530, 3701, 27810, 237151, 2316350, 25135126, 292106400, 3559029501, 45211131460, 600619791201, 8384107777030, 123237338584576, 1904128564485610, 30789744821412401, 518479182191232950, 9057086806410632751, 163745788914416588050
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 03 2022

Keywords

Comments

Shifts 2 places left under 5th-order binomial transform.

Crossrefs

Cf. A000995, A005011, A351028, A351053, A351056, A351128, A351151, A351161.

Programs

  • Mathematica
    nmax = 22; A[] = 0; Do[A[x] = x + x^2 A[x/(1 - 5 x)]/(1 - 5 x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 0; a[1] = 1; a[n_] := a[n] = Sum[Binomial[n - 2, k] 5^k a[n - k - 2], {k, 0, n - 2}]; Table[a[n], {n, 0, 22}]

Formula

a(0) = 0, a(1) = 1; a(n) = Sum_{k=0..n-2} binomial(n-2,k) * 5^k * a(n-k-2).

A111673 Triangle, generated from A111579.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 5, 3, 1, 1, 1, 15, 11, 4, 1, 1, 1, 52, 49, 19, 5, 1, 1, 1, 203, 257, 109, 29, 6, 1, 1, 1, 877, 1539, 742, 201, 41, 7, 1, 1, 1, 4140, 10299, 5815, 1657, 331, 55, 8, 1, 1, 1, 21147, 75905, 51193, 15821, 3176, 505, 71, 9, 1, 1, 1, 115975, 609441, 498118, 170389, 35451, 5497, 729, 89, 10, 1, 1
Offset: 0

Views

Author

Gary W. Adamson, Aug 14 2005

Keywords

Comments

Columns are inverse binomial transforms of columns (k>0) of A111579.

Examples

			First few rows of the triangle are:
  1,
  1, 1,
  1, 1, 1,
  1, 2, 1, 1,
  1, 5, 3, 1, 1,
  1, 15, 11, 4, 1, 1,
  1, 52, 49, 19, 5, 1, 1,
  1, 203, 257, 109, 29, 6, 1, 1,
  1, 877, 1539, 742, 201, 41, 7, 1, 1,
  1, 4140, 10299, 5815, 1657, 331, 55, 8, 1, 1,
  ...
Inverse binomial transform of column 2 of A111579 (1, 2, 5, 15, 52, 203...) = column 2 (1, 1, 2, 5, 15, 52...).

Links

Crossrefs

Cf. A111579, A008277.
For two other versions of this triangle see A241578, A241579.
Rows and columns give A000110, A004211, A004212, A004213, A005011, A005012, A028387, A241577.

Extensions

More terms from N. J. A. Sloane, Apr 29 2014
Previous Showing 11-20 of 21 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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