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 11-20 of 28 results. Next

A075508 Shifts one place left under 9th-order binomial transform.

Original entry on oeis.org

1, 1, 10, 109, 1351, 19612, 333451, 6493069, 141264820, 3376695763, 87799365343, 2465959810690, 74353064138749, 2393123710957813, 81812390963020066, 2958191064076428793, 112727516544416978299, 4513118224822056822772, 189305466502867876489519
Offset: 0

Views

Author

Wolfdieter Lang, Oct 02 2002

Keywords

Comments

Previous name was: Row sums of triangle A075504 (for n>=1).

Links

Crossrefs

Shifts one place left under k-th order binomial transform, k=1..10: A000110, A004211, A004212, A004213, A005011, A005012, A075506, A075507, A075508, A075509.

Programs

  • GAP
    List([0..20],n->Sum([0..n],m->9^(n-m)*Stirling2(n,m))); # Muniru A Asiru, Mar 20 2018
  • Maple
    [seq(factorial(k)*coeftayl(exp((exp(9*x)-1)/9), x = 0, k), k=0..20)]; # Muniru A Asiru, Mar 20 2018
  • Mathematica
    Table[9^n BellB[n, 1/9], {n, 0, 20}] (* Vladimir Reshetnikov, Oct 20 2015 *)

Formula

a(n) = Sum_{m=0..n} 9^(n-m)*S2(n,m), with S2(n,m) = A008277(n,m) (Stirling2).
E.g.f.: exp((exp(9*x)-1)/9).
O.g.f.: Sum_{k>=0} x^k/Product_{j=1..k} (1 - 9*j*x). - Ilya Gutkovskiy, Mar 20 2018
a(n) ~ 9^n * n^n * exp(n/LambertW(9*n) - 1/9 - n) / (sqrt(1 + LambertW(9*n)) * LambertW(9*n)^n). - Vaclav Kotesovec, Jul 15 2021

Extensions

a(0)=1 inserted and new name by Vladimir Reshetnikov, Oct 20 2015

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

A351049 G.f. A(x) satisfies: A(x) = 1 + x + x^2 * A(x/(1 - 3*x)) / (1 - 3*x).

Original entry on oeis.org

1, 1, 1, 4, 16, 67, 307, 1585, 9235, 59548, 415564, 3094807, 24452785, 204611653, 1810429597, 16892405896, 165592138372, 1698918207403, 18184602679435, 202577753111653, 2344503929765023, 28146188358379120, 349996346545057288, 4501360727764475503
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 30 2022

Keywords

Comments

Shifts 2 places left under 3rd-order binomial transform.

Links

Crossrefs

Cf. A004212, A007472, A007476, A351050.

Programs

  • Mathematica
    nmax = 23; A[] = 0; Do[A[x] = 1 + x + x^2 A[x/(1 - 3 x)]/(1 - 3 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] 3^k a[n - k - 2], {k, 0, n - 2}]; Table[a[n], {n, 0, 23}]

Formula

a(0) = a(1) = 1; a(n) = Sum_{k=0..n-2} binomial(n-2,k) * 3^k * a(n-k-2).

A337039 a(n) = exp(-1/3) * Sum_{k>=0} (3*k - 1)^n / (3^k * k!).

Original entry on oeis.org

1, 0, 3, 9, 54, 351, 2673, 22842, 216513, 2248965, 25351704, 307699965, 3995419365, 55207193328, 808078734999, 12480510487509, 202697232446070, 3451417004044323, 61450890989472837, 1141331486235356178, 22066085726516137149, 443236553318792110113, 9233934519951699602400
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 12 2020

Keywords

Crossrefs

Cf. A000296, A003575, A004212, A337038, A337040, A337041, A337042, A337043.

Programs

  • Mathematica
    nmax = 22; CoefficientList[Series[Exp[(Exp[3 x] - 1)/3 - x], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k] 3^k a[n - k - 1], {k, 1, n - 1}]; Table[a[n], {n, 0, 22}]
Table[Sum[(-1)^(n - k) Binomial[n, k] 3^k BellB[k, 1/3], {k, 0, n}], {n, 0, 22}]

Formula

G.f. A(x) satisfies: A(x) = (1 - 3*x + x*A(x/(1 - 3*x))) / (1 - 2*x - 3*x^2).
G.f.: (1/(1 + x)) * Sum_{k>=0} (x/(1 + x))^k / Product_{j=1..k} (1 - 3*j*x/(1 + x)).
E.g.f.: exp((exp(3*x) - 1) / 3 - x).
a(0) = 1; a(n) = Sum_{k=1..n-1} binomial(n-1,k) * 3^k * a(n-k-1).
a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * A004212(k).
a(n) ~ 3^(n - 1/3) * n^(n - 1/3) * exp(n/LambertW(3*n) - n - 1/3) / (sqrt(1 + LambertW(3*n)) * LambertW(3*n)^(n - 1/3)). - Vaclav Kotesovec, Jun 26 2022

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:

A337593 a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} binomial(n,k)^2 * k * 3^(k-1) * a(n-k).

Original entry on oeis.org

1, 1, 5, 42, 573, 11226, 294804, 9946791, 417064365, 21187915362, 1278636342660, 90195692894451, 7338668846348844, 681052861293535251, 71405270562056271741, 8388541745045127600597, 1096298129481068449931085, 158383969954582566159384786, 25153555538082783169267336764
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 02 2020

Keywords

Crossrefs

Cf. A004212, A337592, A337594, A337595, A337597.

Programs

  • Mathematica
    a[0] = 1; a[n_] := a[n] = (1/n) Sum[Binomial[n, k]^2 k 3^(k - 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]
nmax = 18; CoefficientList[Series[Exp[(BesselI[0, 2 Sqrt[3 x]] - 1)/3], {x, 0, nmax}], x] Range[0, nmax]!^2

Formula

Sum_{n>=0} a(n) * x^n / (n!)^2 = exp((BesselI(0,2*sqrt(3*x)) - 1) / 3).
Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(Sum_{n>=1} 3^(n-1) * x^n / (n!)^2).

A351144 G.f. A(x) satisfies: A(x) = 1 + x^2 * A(x/(1 - 3*x)) / (1 - 3*x).

Original entry on oeis.org

1, 0, 1, 3, 10, 39, 181, 972, 5797, 37389, 258202, 1905681, 15016465, 125920872, 1117950913, 10452866439, 102485649754, 1050464300187, 11231883627301, 125055844922916, 1447371528438565, 17382103226123313, 216221862096537994, 2781342531957176085, 36942930754308211969
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 02 2022

Keywords

Comments

Shifts 2 places left under 3rd-order binomial transform.

Crossrefs

Cf. A000994, A004212, A351049, A351143, A351150, A351151, A351152.

Programs

  • Mathematica
    nmax = 24; A[] = 0; Do[A[x] = 1 + x^2 A[x/(1 - 3 x)]/(1 - 3 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] 3^k a[n - k - 2], {k, 0, n - 2}]; Table[a[n], {n, 0, 24}]

Formula

a(0) = 1, a(1) = 0; a(n) = Sum_{k=0..n-2} binomial(n-2,k) * 3^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:

A351053 G.f. A(x) satisfies: A(x) = x + x^2 * A(x/(1 - 3*x)) / (1 - 3*x).

Original entry on oeis.org

0, 1, 0, 1, 6, 28, 126, 613, 3438, 22159, 157362, 1189126, 9436320, 78690781, 692478684, 6439539457, 63106488618, 648453907216, 6952719052134, 77521908188737, 897132401326458, 10764085132255807, 133774484448519294, 1720018195807299418, 22847325911461934352
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 03 2022

Keywords

Comments

Shifts 2 places left under 3rd-order binomial transform.

Crossrefs

Cf. A000995, A004212, A351028, A351049, A351128, A351132, A351144, A351161.

Programs

  • Mathematica
    nmax = 24; A[] = 0; Do[A[x] = x + x^2 A[x/(1 - 3 x)]/(1 - 3 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] 3^k a[n - k - 2], {k, 0, n - 2}]; Table[a[n], {n, 0, 24}]

Formula

a(0) = 0, a(1) = 1; a(n) = Sum_{k=0..n-2} binomial(n-2,k) * 3^k * a(n-k-2).

A351757 G.f. A(x) satisfies: A(x) = 1 + x * A(x/(1 - 3*x)) / (1 - 3*x)^2.

Original entry on oeis.org

1, 1, 7, 43, 289, 2239, 19699, 192025, 2042971, 23520715, 291099349, 3849621019, 54110928355, 804827487493, 12619011606775, 207885167529523, 3587864566792753, 64705561315720135, 1216574535057705979, 23797327657083197113, 483390249416359706995
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 18 2022

Keywords

Crossrefs

Cf. A004212, A040027, A255927, A351756.

Programs

  • Mathematica
    nmax = 20; A[] = 0; Do[A[x] = 1 + x A[x/(1 - 3 x)]/(1 - 3 x)^2 + O[x]^(nmax + 1) // Normal,nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k - 1] 3^(k - 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 20}]

Formula

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k-1) * 3^(k-1) * a(n-k).
Previous Showing 11-20 of 28 results. Next

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

Content is available under The OEIS End-User License Agreement.