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 71-78 of 78 results.

A319110 Expansion of Product_{k>=1} 1/(1 - (k - 1)*x^k).

Original entry on oeis.org

1, 0, 1, 2, 4, 6, 13, 18, 37, 56, 101, 152, 285, 410, 713, 1118, 1830, 2780, 4618, 6934, 11278, 17092, 26894, 40822, 64435, 96372, 149299, 225104, 345131, 515394, 788176, 1169962, 1772957, 2632458, 3950365, 5849260, 8748993, 12867848, 19135894, 28126614, 41598695
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 10 2018

Keywords

Links

Crossrefs

Cf. A006906, A034008, A052847, A074141, A267007.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0 or i=1,
       0^n, b(n, i-1)+(i-1)*b(n-i, min(n-i, i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..42);  # Alois P. Heinz, Aug 19 2019
  • Mathematica
    nmax = 40; CoefficientList[Series[Product[1/(1 - (k - 1) x^k), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 40; CoefficientList[Series[Exp[Sum[Sum[(j - 1)^k x^(j k)/k, {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d (d - 1)^(k/d), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 40}]

Formula

G.f.: exp(Sum_{k>=1} Sum_{j>=1} (j - 1)^k*x^(j*k)/k).
From Vaclav Kotesovec, Sep 11 2018: (Start)
a(n) ~ c * 2^(2*n/5), where
c = 28108804.248904780960402246466460350520790117596512766842168... if mod(n,5) = 0
c = 28108804.010850549080284030388905319123062152339902207992657... if mod(n,5) = 1
c = 28108804.067769166625741650205643600577757560110636366636106... if mod(n,5) = 2
c = 28108804.083581827971851596540314974909801290757084687583764... if mod(n,5) = 3
c = 28108804.058853893104368046896759214442695016905368229405793... if mod(n,5) = 4
(End)

A319756 Expansion of Product_{k>=1} (1 - x^k)/(1 - k*x^k).

Original entry on oeis.org

1, 0, 1, 2, 5, 6, 18, 20, 52, 76, 151, 214, 486, 638, 1265, 1990, 3572, 5288, 9968, 14568, 26270, 40246, 68326, 104414, 182191, 271892, 457636, 708012, 1164554, 1774422, 2945077, 4450020, 7261298, 11138514, 17827308, 27228060, 43860232, 66305840, 105486224, 161284674, 253846152
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 27 2018

Keywords

Comments

Convolution of A006906 and A010815.

Links

Crossrefs

Cf. A006906, A010815, A267005.

Programs

  • Maple
    a:=series(mul((1-x^k)/(1-k*x^k),k=1..100),x=0,41): seq(coeff(a,x,n),n=0..40); # Paolo P. Lava, Apr 02 2019
  • Mathematica
    nmax = 40; CoefficientList[Series[Product[(1 - x^k)/(1 - k*x^k), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 40; CoefficientList[Series[Exp[Sum[Sum[d (d^(k/d) - 1), {d, Divisors[k]}] x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d (d^(k/d) - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 40}]
  • PARI
    x='x+O('x^40); Vec(prod(n=1, 40, (1-x^n)/(1-n*x^n))) \\ Altug Alkan, Sep 27 2018

Formula

G.f.: exp(Sum_{k>=1} ( Sum_{d|k} d*(d^(k/d) - 1) ) * x^k/k).
From Vaclav Kotesovec, Sep 27 2018: (Start)
a(n) ~ c * 3^(n/3), where
c = 4613.026226899587659466790384528262900057997961519... if mod(n,3)=0
c = 4612.491093385908314202944836907761153110706939289... if mod(n,3)=1
c = 4612.543916007416515763773288072302642108310934844... if mod(n,3)=2
(End)

A320652 Expansion of 1/(2 - Product_{k>=1} 1/(1 - k*x^k)).

Original entry on oeis.org

1, 1, 4, 13, 45, 147, 497, 1643, 5490, 18252, 60812, 202364, 673915, 2243295, 7468973, 24865272, 82783967, 275605513, 917563193, 3054785032, 10170143277, 33858882922, 112724577088, 375287739083, 1249425198725, 4159643200494, 13848474406054, 46104972636634, 153494780854254
Offset: 0

Views

Author

Ilya Gutkovskiy, Oct 18 2018

Keywords

Comments

Invert transform of A006906.

Links

Crossrefs

Cf. A006906, A055887, A257674, A299162.

Programs

  • Maple
    a:=series(1/(2-mul(1/(1-k*x^k),k=1..100)),x=0,29): seq(coeff(a,x,n),n=0..28); # Paolo P. Lava, Apr 02 2019
  • Mathematica
    nmax = 28; CoefficientList[Series[1/(2 - Product[1/(1 - k x^k), {k, 1, nmax}]), {x, 0, nmax}], x]
nmax = 28; CoefficientList[Series[1/(1 - Sum[k x^k/Product[(1 - j x^j), {j, 1, k}], {k, 1, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[Total[Times@@@IntegerPartitions[k]] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 28}]

Formula

G.f.: 1/(1 - Sum_{k>=1} k*x^k / Product_{j=1..k} (1 - j*x^j)).
a(0) = 1; a(n) = Sum_{k=1..n} A006906(k)*a(n-k).

A325536 Sum of sums of omegas of parts over all integer partitions of n.

Original entry on oeis.org

0, 0, 1, 2, 6, 9, 19, 28, 51, 75, 119, 170, 261, 362, 525, 723, 1019, 1373, 1890, 2512, 3386, 4452, 5893, 7658, 10017, 12881, 16627, 21210, 27097, 34266, 43392, 54462, 68399, 85285, 106305, 131712, 163132, 200936, 247332, 303066, 370989, 452296, 550875, 668495
Offset: 0

Views

Author

Gus Wiseman, May 08 2019

Keywords

Comments

Also omega of the product of products of parts over all integer partitions of n.
The omega of n is A001222(n), the number of prime factors of n counted with multiplicity.

Examples

			The integer partitions of 5 are {(5), (4,1), (3,2), (3,1,1), (2,2,1), (2,1,1,1), (1,1,1,1,1)} with products {5,4,6,3,4,2,1} with product 2880 with omega 9, so a(5) = 9.

Crossrefs

Cf. A001222, A006128, A006906, A007870, A066186, A066633, A145519, A215366, A325507.

Programs

  • Mathematica
    Table[Plus@@PrimeOmega/@Join@@IntegerPartitions[n],{n,0,30}]

Formula

a(n) = A001222(A007870(n)).

A356524 Expansion of e.g.f. Product_{k>0} 1/(1 - k * x^k)^(1/k!).

Original entry on oeis.org

1, 1, 4, 15, 100, 565, 5946, 46039, 605256, 6646329, 103614490, 1320840631, 27185208876, 401901829069, 9042437722878, 168984439301175, 4257225193170256, 85582303577644465, 2593970612953642386, 57441717948059605927, 1862688382990615542900
Offset: 0

Views

Author

Seiichi Manyama, Aug 10 2022

Keywords

Crossrefs

Cf. A006906, A209902, A294462, A354849, A356487.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/prod(k=1, N, (1-k*x^k)^(1/k!))))
  • PARI
    a354849(n) = (n-1)!*sumdiv(n, d, d^(n/d)/(d-1)!);
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, a354849(j)*binomial(i-1, j-1)*v[i-j+1])); v;

Formula

a(0) = 1; a(n) = Sum_{k=1..n} A354849(k) * binomial(n-1,k-1) * a(n-k).

A368091 Triangle read by rows. T(n, k) = Sum_{p in P(n, k)} Product_{r in p} r, where P(n, k) are the partitions of n with length k.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 3, 2, 1, 0, 4, 7, 2, 1, 0, 5, 10, 7, 2, 1, 0, 6, 22, 18, 7, 2, 1, 0, 7, 28, 34, 18, 7, 2, 1, 0, 8, 50, 62, 50, 18, 7, 2, 1, 0, 9, 60, 121, 86, 50, 18, 7, 2, 1, 0, 10, 95, 182, 189, 118, 50, 18, 7, 2, 1
Offset: 0

Views

Author

Peter Luschny, Dec 11 2023

Keywords

Examples

			Table T(n, k) starts:
  [0] [1]
  [1] [0, 1]
  [2] [0, 2,  1]
  [3] [0, 3,  2,   1]
  [4] [0, 4,  7,   2,  1]
  [5] [0, 5, 10,   7,  2,  1]
  [6] [0, 6, 22,  18,  7,  2,  1]
  [7] [0, 7, 28,  34, 18,  7,  2, 1]
  [8] [0, 8, 50,  62, 50, 18,  7, 2, 1]
  [9] [0, 9, 60, 121, 86, 50, 18, 7, 2, 1]

Crossrefs

Cf. A368090, A074141, A023855, A006906 (row sums).

Programs

  • SageMath
    def T(n, k):
    return sum(product(r for r in p) for p in Partitions(n, length=k))
for n in range(10): print([T(n, k) for k in range(n + 1)])

A111377 Triangle of sum of product of partitions of n where largest part is k.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 6, 3, 4, 1, 6, 9, 4, 5, 1, 14, 18, 12, 5, 6, 1, 14, 30, 24, 15, 6, 7, 1, 30, 48, 56, 30, 18, 7, 8, 1, 30, 99, 80, 70, 36, 21, 8, 9, 1, 62, 135, 180, 125, 84, 42, 24, 9, 10, 1, 62, 237, 276, 250, 150, 98, 48, 27, 10, 11, 1, 126, 390, 540, 420, 336, 175, 112, 54, 30
Offset: 1

Views

Author

Henry Bottomley, Nov 09 2005

Keywords

Examples

			Triangle starts 1; 1,2; 1,2,3; 1,6,3,4; 1,6,9,4,5; 1,14,18,12,5,6; etc.
T(6,3)=18 since partitions of 6 with largest part 3 is 3+3, 3+2+1, 3+1+1+1 and 3*3 + 3*2*1 + 3*1*1*1 = 18.

Crossrefs

Row sums are A006906.

Formula

T(n, k)=k*sum_j {0<=j<=k} a(n-k, j) starting with T(0, 0)=1; T(n, n)=n for n>=1.

A300411 a(0) = a(1) = 1; a(n) = [x^n] Product_{k=1..n-1} 1/(1 - a(k)*x^a(k)).

Original entry on oeis.org

1, 1, 1, 4, 9, 14, 19, 24, 45, 75, 105, 135, 229, 359, 503, 647, 1047, 1591, 2272, 2972, 4696, 6996, 9844, 12894, 20064, 29538, 41204, 54407, 84457, 123723, 171757, 225939, 348643, 508693, 703815, 923529, 1423892, 2076942, 2870977, 3763380, 5778379, 8414332, 11621307
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 05 2018

Keywords

Crossrefs

Cf. A006906, A093637, A151945, A196545, A291698, A293806, A293807, A296387.

Programs

  • Mathematica
    a[n_] := a[n] = SeriesCoefficient[Product[1/(1 - a[k] x^a[k]), {k, 1, n - 1}], {x, 0, n}]; a[0] = a[1] = 1; Table[a[n], {n, 0, 42}]

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies A(x) = -x - 2*x^2 + Product_{n>=1} 1/(1 - a(n)*x^a(n)).
Previous Showing 71-78 of 78 results.

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

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