A147559 -id:A147559 - cp's OEIS frontend

A147654 Result of using the positive integers 1,2,3,... as coefficients in an infinite polynomial series in x and then expressing this series as Product_{k>=1} (1+a(k)x^k).

1, 2, 1, 3, 0, -2, 0, 9, 0, -6, 0, 4, 0, -18, 0, 93, 0, -54, 0, 72, 0, -186, 0, 232, 0, -630, 0, 1020, 0, -2106, 0, 10881, 0, -7710, 0, 13824, 0, -27594, 0, 49440, 0, -97902, 0, 191844, 0, -364722, 0, 590800, 0, -1340622, 0, 2656920, 0, -4918482, 0, 9791784, 0, -18512790

Offset: 1

Neil Fernandez, Nov 09 2008

sign

			From the positive integers 1,2,3,..., construct the series 1+x+2x^2+3x^3+4x^4+... a(1) is always the coefficient of x, here 1. Divide by (1+a(1)x), i.e. here (1+x), to get the quotient (1+a(2)x^2+...), which here gives a(2)=2. Then divide this quotient by (1+a(2)x^2), i.e. here (1+2x^2), to get (1+a(3)x^3+...), giving a(3)=1.

Seiichi Manyama, Table of n, a(n) for n = 1..500

Cf. A028310, A147541, A147559.

Product_{k>=1} (1+a(k)*x^k) = 1 + Sum_{k>=1} k*x^k. - Seiichi Manyama, Jun 24 2018

More terms from Seiichi Manyama, Jun 23 2018

A147880 Expansion of Product_{k > 0} (1 + A005229(k)*x^k).

Original entry on oeis.org

1, 1, 1, 3, 5, 8, 12, 21, 30, 50, 75, 110, 169, 249, 361, 539, 757, 1076, 1583, 2207, 3121, 4415, 6184, 8468, 11775, 16274, 22314, 30601, 41745, 56412, 77008, 103507, 138383, 186928, 249855, 333375, 443898, 588402, 778276, 1031126, 1356945, 1780645

Offset: 0

Roger L. Bagula, Nov 16 2008

nonn

			From _Petros Hadjicostas_, Apr 10 2020: (Start)
Let f(m) = A005229(m). Using the strict partitions of each n (see A000009), we get
a(1) = f(1) = 1,
a(2) = f(2) = 1,
a(3) = f(3) + f(1)*f(2) = 2 + 1*1 = 3,
a(4) = f(4) + f(1)*f(3) = 3 + 1*2 = 5,
a(5) = f(5) + f(1)*f(4) + f(2)*f(3) = 3 + 1*3 + 1*2 = 8,
a(6) = f(6) + f(1)*f(5) + f(2)*f(4) + f(1)*f(2)*f(3) = 4 + 1*3 + 1*3 + 1*1*2 = 12,
a(7) = f(7) + f(1)*f(6) + f(2)*f(5) + f(3)*f(4) + f(1)*f(2)*f(4) = 5 + 1*4 + 1*3 + 2*3 + 1*1*3 = 21. (End)

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Cf. A000009, A004001, A005229, A147559, A147654, A147655, A147665.

(*A005229*) f[n_Integer?Positive] := f[n] = f[ f[n - 2]] + f[n - f[n - 2]]; f[0] = 0; f[1] = f[2] = 1;
P[x_, n_] := P[x, n] = Product[1 + f[m] *x^m, {m, 0, n}];
Take[CoefficientList[P[x, 45], x], 45] (* Program simplified by Petros Hadjicostas, Apr 13 2020 *)

\\ here B(n) is A005229 as vector.
B(n)={my(a=vector(n, i, 1)); for(n=3, n, a[n] = a[a[n-2]] + a[n-a[n-2]]); a}
seq(n)={my(v=B(n)); Vec(prod(k=1, n, 1 + v[k]*x^k + O(x*x^n)))} \\ Andrew Howroyd, Apr 10 2020

G.f.: Product_{k > 0} (1 + A005229(k)*x^k).

a(n) = Sum_{(b_1,...,b_n)} f(1)^b_1 * f(2)^b_2 * ... * f(n)^b_n, where f(m) = A005229(m), and the sum is taken over all lists (b_1,...,b_n) with b_j in {0,1} and Sum_{j=1..n} j*b_j = n.

Various sections edited by Joerg Arndt and Petros Hadjicostas, Apr 10 2020

A147871 Expansion of Product_{k > 0} (1 + A147665(k)*x^k).

Original entry on oeis.org

1, 1, 1, 3, 4, 7, 10, 15, 24, 37, 49, 73, 105, 142, 208, 294, 391, 538, 752, 988, 1359, 1812, 2410, 3232, 4270, 5598, 7454, 9721, 12639, 16625, 21445, 27649, 35793, 46235, 59141, 76215, 96975, 123262, 157671, 199625, 252591, 319792, 403262, 507682

Offset: 0

Roger L. Bagula, Nov 16 2008

nonn

			From _Petros Hadjicostas_, Apr 11 2020: (Start)
Let f(m) = A147665(m). Using the strict partitions of each n (see A000009), we get
a(1) = f(1) = 1,
a(2) = f(2) = 1,
a(3) = f(3) + f(1)*f(2) = 2 + 1*1 = 3,
a(4) = f(4) + f(1)*f(3) = 2 + 1*2 = 4,
a(5) = f(5) + f(1)*f(4) + f(2)*f(3) = 3 + 1*2 + 1*2 = 7,
a(6) = f(6) + f(1)*f(5) + f(2)*f(4) + f(1)*f(2)*f(3) = 3 + 1*3 + 1*2 + 1*1*2 = 10,
a(7) = f(7) + f(1)*f(6) + f(2)*f(5) + f(3)*f(4) + f(1)*f(2)*f(4) = 3 + 1*3 + 1*3 + 2*2 + 1*1*2 = 15. (End)

Cf. A000009, A004001, A147559, A147654, A147655, A147665, A147880.

(*A147665*) f[0] = 0; f[1] = 1; f[2] = 1; f[n_] := f[n] = f[f[n - 1]] + If[Mod[ n, 3] == 0, f[f[n/3]], If[Mod[n, 3] == 1, f[f[(n - 1)/3]], f[n - f[(n - 2)/3]]]];
P[x_, n_] := P[x, n] = Product[1 + f[m]*x^m, {m, 0, n}];
Take[CoefficientList[P[x, 45], x], 45] (* Program simplified by Petros Hadjicostas, Apr 13 2020 *)

a(n) = [x^n] Product_{k > 0} (1 + A147665(k)*x^k).

a(n) = Sum_{(b_1,...,b_n)} f(1)^b_1 * f(2)^b_2 * ... * f(n)^b_n, where f(m) = A147665(m), and the sum is taken over all lists (b_1,...,b_n) with b_j in {0,1} and Sum_{j=1..n} j*b_j = n. - Petros Hadjicostas, Apr 11 2020

Various sections edited by Petros Hadjicostas, Apr 11 2020

A147869 Expansion of Product_{k>0} (1 + A004001(k)*x^k).

Original entry on oeis.org

1, 1, 1, 3, 4, 7, 11, 17, 25, 41, 59, 86, 125, 180, 263, 382, 536, 738, 1073, 1466, 2028, 2841, 3889, 5275, 7211, 9800, 13249, 17860, 23948, 31921, 42864, 56802, 75115, 99788, 131239, 172870, 226789, 296404, 386745, 504939, 655227, 849628, 1101270

Offset: 0

Roger L. Bagula, Nov 16 2008

nonn

			From _Petros Hadjicostas_, Apr 11 2020: (Start)
Let f(m) = A004001(m). Using the strict partitions of each n (see A000009), we get
a(1) = f(1) = 1,
a(2) = f(2) = 1,
a(3) = f(3) + f(1)*f(2) = 2 + 1*1 = 3,
a(4) = f(4) + f(1)*f(3) = 2 + 1*2 = 4,
a(5) = f(5) + f(1)*f(4) + f(2)*f(3) = 3 + 1*2 + 1*2 = 7,
a(6) = f(6) + f(1)*f(5) + f(2)*f(4) + f(1)*f(2)*f(3) = 4 + 1*3 + 1*2 + 1*1*2 = 11,
a(7) = f(7) + f(1)*f(6) + f(2)*f(5) + f(3)*f(4) + f(1)*f(2)*f(4) = 4 + 1*4 + 1*3 + 2*2 + 1*1*2 = 17. (End)

Cf. A000009, A004001, A147654, A147655, A147559, A147880.

f[0] = 0; f[1] = 1; f[2] = 1; f[n_] := f[n] = f[f[n - 1]] + f[n - f[n - 1]];
P[x_, n_] := P[x, n] = Product[1 + f[m]*x^m, {m, 0, n}];
Take[CoefficientList[P[x, 45], x], 45]

a(n) = [x^n] Product_{k > 0} (1 + A004001(k)*x^k).

a(n) = Sum_{(b_1,...,b_n)} f(1)^b_1 * f(2)^b_2 * ... * f(n)^b_n, where f(m) = A004001(m), and the sum is taken over all lists (b_1,...,b_n) with b_j in {0,1} and Sum_{j=1..n} j*b_j = n. - Petros Hadjicostas, Apr 11 2020

Various sections edited by Petros Hadjicostas, Apr 11 2020

A316083 Product_{k>=1} (1 + a(k)x^k) = Sum_{k>=0} (kx)^k.

Original entry on oeis.org

1, 4, 23, 233, 2800, 42832, 763220, 15761709, 366711200, 9537738596, 273549419552, 8587897407548, 292755986184548, 10773140836162944, 425587704331945152, 17966341563465800813, 807152054953801845760, 38451432814472749509872, 1936082850634342992601636

Offset: 1

Seiichi Manyama, Jun 23 2018

nonn

			(1+x)*(1+4*x^2)*(1+23*x^3)*(1+233*x^4)* ... = 1 + x + 4*x^2 + 27*x^3 + 256*x^4 + ... .

Seiichi Manyama, Table of n, a(n) for n = 1..386

Cf. A000312, A147559, A147654, A158094, A316085.

a(n) ~ n^n. - Vaclav Kotesovec, Jun 18 2019

A316086 Product_{k>=1} 1/(1 - a(k)x^k) = 1 + Sum_{k>=1} k^2x^k.

Original entry on oeis.org

1, 3, 5, -2, -6, -35, -4, 8, 16, -62, -158, -1149, 56, 556, 680, -1566, -4966, 3544, 6132, 20268, 15792, -75194, -162814, -1246153, 333904, 805112, 235968, -2775034, -5259270, 7160120, 15199212, 25457976, -4403208, -119142438, -161039518, 211525304, 635914664

Offset: 1

Seiichi Manyama, Jun 24 2018

sign

			1/((1-x)*(1-3*x^2)*(1-5*x^3)*(1+2*x^4)* ... ) = 1 + x + 4*x^2 + 9*x^3 + 16*x^4 + ... .

Seiichi Manyama, Table of n, a(n) for n = 1..500

Cf. A147559, A253909, A316087.

A359265 Product_{n>=1} (1 + a(n) * x^n) = 1 + Sum_{n>=1} n^3 * x^n.

Original entry on oeis.org

1, 8, 19, 45, -72, -224, -72, 3465, 1656, -4752, -31248, -440, 62064, 415008, 936432, 6776793, -16454232, -24983784, 74804904, 468856296, 236519784, -2495390904, -8714625696, -8228470832, 62274531168, 155889061848, -47291852448, -1334769988176, -4304113760232

Offset: 1

Seiichi Manyama, Dec 28 2022

sign

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Cf. A000578, A147559, A147654, A316083.

S:= 1 + x*(x^2 + 4*x + 1)/(x - 1)^4:
for n from 1 to 30 do
  SS:= series(S,x,n+1);
  A[n]:= coeff(SS,x,n);
  S:= S/(1+A[n]*x^n);
od:
seq(A[i],i=1..30); # Robert Israel, Dec 28 2022

A359407 Product_{n>=1} (1 + a(n) * x^n) = 1 + Sum_{n>=1} (n * (n + 1) / 2) * x^n.

Original entry on oeis.org

1, 3, 3, 7, -1, -8, -3, 48, 1, -39, -6, 34, -25, -293, 24, 2649, -35, -2104, -239, 3387, 323, -15322, -182, 27551, -2477, -121985, 3663, 275053, -360, -942231, -26801, 8245538, 40690, -7506845, 11479, 19149901, -297030, -61205072, 442870, 158934665, 284075, -490800388, -3338212

Offset: 1

Seiichi Manyama, Dec 30 2022

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Cf. A000217, A147559.

A147879 Expansion of Product_{k>=1} (1 + x^k*A005185(k)).

Original entry on oeis.org

1, 1, 1, 3, 5, 8, 12, 21, 29, 49, 73, 105, 162, 236, 338, 502, 706, 984, 1441, 1998, 2800, 3934, 5472, 7407, 10210, 14053, 19066, 25986, 35134, 47010, 63739, 85008, 112610, 150861, 200133, 264838, 349587, 459970, 602763, 792220, 1034136, 1345530

Offset: 0

Roger L. Bagula, Nov 16 2008

nonn

Cf. A004001, A005185, A147559, A147654, A147655, A147665, A147869, A147871.

f[n_Integer?Positive] := f[n] = f[n - f[n - 1]] + f[n - f[n - 2]]; f[0] = 0; f[1] = f[2] = 1; (* A005185 *)
nmax = 41; CoefficientList[Series[Product[(1 + f[k] * x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Georg Fischer, Dec 10 2020 *)

\\ here B(n) is A005185 as vector.
B(n)={my(A=vector(n, k, 1)); for(k=3, n, A[k]= A[k-A[k-1]]+ A[k-A[k-2]]); A}
seq(n)=my(v=B(n)); {Vec(prod(k=1, #v, 1 + x^k*v[k] + O(x*x^n)))} \\ Andrew Howroyd, Dec 10 2020

Definition corrected by Georg Fischer, Dec 10 2020

cp's OEIS Frontend

A147654 Result of using the positive integers 1,2,3,... as coefficients in an infinite polynomial series in x and then expressing this series as Product_{k>=1} (1+a(k)x^k).

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

Extensions

A147880 Expansion of Product_{k > 0} (1 + A005229(k)*x^k).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A147871 Expansion of Product_{k > 0} (1 + A147665(k)*x^k).

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

Extensions

A147869 Expansion of Product_{k>0} (1 + A004001(k)*x^k).

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

Extensions

A316083 Product_{k>=1} (1 + a(k)*x^k) = Sum_{k>=0} (k*x)^k.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A316086 Product_{k>=1} 1/(1 - a(k)*x^k) = 1 + Sum_{k>=1} k^2*x^k.

Views

Author

Keywords

Examples

Links

Crossrefs

A359265 Product_{n>=1} (1 + a(n) * x^n) = 1 + Sum_{n>=1} n^3 * x^n.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

A359407 Product_{n>=1} (1 + a(n) * x^n) = 1 + Sum_{n>=1} (n * (n + 1) / 2) * x^n.

Views

Author

Keywords

Links

Crossrefs

A147879 Expansion of Product_{k>=1} (1 + x^k*A005185(k)).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Extensions

A316083 Product_{k>=1} (1 + a(k)x^k) = Sum_{k>=0} (kx)^k.

A316086 Product_{k>=1} 1/(1 - a(k)x^k) = 1 + Sum_{k>=1} k^2x^k.