A039924 -id:A039924 - cp's OEIS frontend

A003116 Expansion of the reciprocal of the g.f. defining A039924.

1, 1, 2, 4, 7, 13, 23, 41, 72, 127, 222, 388, 677, 1179, 2052, 3569, 6203, 10778, 18722, 32513, 56455, 98017, 170161, 295389, 512755, 890043, 1544907, 2681554, 4654417, 8078679, 14022089, 24337897, 42242732, 73319574, 127258596, 220878683

Offset: 0

N. J. A. Sloane, Herman P. Robinson

Conjecture: a(n) is the number of compositions p(1) + p(2) + ... + p(m) = n with p(i)-p(i-1) <= 1, see example; cf. A034297. - Vladeta Jovovic, Feb 09 2004
Row sums and central terms of the triangle in A168396: a(n) = A168396(2*n+1,n) and for n > 0: a(n) = Sum_{k=1..n} A168396(n,k). - Reinhard Zumkeller, Sep 13 2013
Former definition was "Expansion of reciprocal of a determinant." - N. J. A. Sloane, Aug 10 2018
From Doron Zeilberger, Aug 10 2018: (Start)
Jovovic's conjecture can be proved as follows. There is a sign-changing involution defined on pairs (L1,L2) where L1 is a partition with difference >= 2 between consecutive parts and L2 is the number of compositions described by Jovovic, with the sign (-1)^(Number of parts of L1).
Let a be the largest part of L1 and b the largest part of L2. If b-a>=2 then move b from L2 to the top of L1, otherwise move a to the top of L2.
Since this is an involution and it changes the sign (the number of parts of L1 changes parity) this proves it, since the g.f. of A039924 is exactly the signed-enumeration of the set given by L1. (End)

			From _Joerg Arndt_, Dec 29 2012: (Start)
There are a(6)=23 compositions p(1)+p(2)+...+p(m)=6 such that p(k)-p(k-1) <= 1:
[ 1]  [ 1 1 1 1 1 1 ]
[ 2]  [ 1 1 1 1 2 ]
[ 3]  [ 1 1 1 2 1 ]
[ 4]  [ 1 1 2 1 1 ]
[ 5]  [ 1 1 2 2 ]
[ 6]  [ 1 2 1 1 1 ]
[ 7]  [ 1 2 1 2 ]
[ 8]  [ 1 2 2 1 ]
[ 9]  [ 1 2 3 ]
[10]  [ 2 1 1 1 1 ]
[11]  [ 2 1 1 2 ]
[12]  [ 2 1 2 1 ]
[13]  [ 2 2 1 1 ]
[14]  [ 2 2 2 ]
[15]  [ 2 3 1 ]
[16]  [ 3 1 1 1 ]
[17]  [ 3 1 2 ]
[18]  [ 3 2 1 ]
[19]  [ 3 3 ]
[20]  [ 4 1 1 ]
[21]  [ 4 2 ]
[22]  [ 5 1 ]
[23]  [ 6 ]
Replacing the condition with p(k)-p(k-1) <= 0 gives integer partitions.
(End)

D. H. Lehmer, Combinatorial and cyclotomic properties of certain tridiagonal matrices. Proceedings of the Fifth Southeastern Conference on Combinatorics, Graph Theory, and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1974), pp. 53-74. Congressus Numerantium, No. X, Utilitas Math., Winnipeg, Man., 1974. MR0441852.
H. P. Robinson, Letter to N. J. A. Sloane, Nov 19 1973.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..4176 (first 401 terms from T. D. Noe)
Roland Bacher, Generic numerical semigroups, hal-03221466 [math.CO], 2021.
George Beck and Shane Chern, Reciprocity between partitions and compositions, arXiv:2108.04363 [math.CO], 2021.
Shalosh B. Ekhad and Doron Zeilberger, D.H. Lehmer's Tridiagonal determinant: An Etude in (Andrews-Inspired) Experimental Mathematics, arXiv:1808.06730 [math.CO], 2018.
Miguel Mendez, Shift-plethysm, Hydra continued fractions, and m-distinct partitions, arXiv:2009.04623 [math.CO], 2020.
Herman P. Robinson, Letter to N. J. A. Sloane, Nov 13 1973.
Herman P. Robinson, Letter to N. J. A. Sloane, Nov 19 1973.

Cf. A003114, A039924, A034297, A168443, A224959.

a003116 n = a168396 (2 * n + 1) n  -- Reinhard Zumkeller, Sep 13 2013

max = 35; f[x_] := 1/Sum[x^k^2*((-1)^k/Product[1 - x^i, {i, 1, k}]), {k, 0, Floor[Sqrt[max]]}]; CoefficientList[ Series[f[x], {x, 0, max}], x](* Jean-François Alcover, Jun 12 2012, after PARI *)
b[n_, k_] := b[n, k] = Expand[If[n == 0, 1, x*
     Sum[b[n - j, j], {j, 1, Min[n, k + 1]}]]];
a[n_] := Total@CoefficientList[b[n, n], x];
Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Apr 14 2022, after Alois P. Heinz in A168443 *)

a(n)=if(n<0,0,polcoeff(1/sum(k=0,sqrtint(n),x^k^2/prod(i=1,k,x^i-1,1+x*O(x^n))),n))

G.f.: 1/(Sum_{k>=0} x^(k^2)(-1)^k/(Product_{i=1..k} 1-x^i)).

a(n) ~ c * d^n, where d = 1/A347901 = 1.73566282453034742565826074971966853... and c = 0.9180565304926754125870866477349969555868577236908640010903420353... - Vaclav Kotesovec, Nov 01 2021

Definition revised by N. J. A. Sloane, Aug 10 2018 at the suggestion of Doron Zeilberger

A224898 G.f.: Sum_{n>=0} (-1)^n* x^(n*(n+1)) / Product_{k=1..n} (1-x^k).

Original entry on oeis.org

1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 2, 2, 2, 2, 2, 1, 1, 0, -1, -2, -2, -4, -4, -5, -5, -6, -5, -6, -4, -4, -3, -2, 1, 1, 4, 5, 8, 9, 12, 12, 15, 15, 17, 16, 18, 15, 16, 13, 13, 8, 7, 1, 0, -7, -9, -17, -19, -27, -29, -37, -38, -46, -46, -53, -51, -57, -53, -57, -51, -53, -45, -45, -32, -31

Offset: 0

Paul D. Hanna, Jul 24 2013

Conjecture: a(n+1) = A286744(n) - A286745(n). - George Beck May 13 2017

			G.f.: A(x) = 1 - x^2 - x^3 - x^4 - x^5 + x^8 + x^9 + 2*x^10 + 2*x^11 + 2*x^12 + 2*x^13 + 2*x^14 + x^15 + x^16 - x^18 +...
where
A(x) = 1 - x^2/(1-x) + x^6/((1-x)*(1-x^2)) - x^12/((1-x)*(1-x^2)*(1-x^3)) + x^20/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)) - x^30/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)*(1-x^5)) +...

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1001 terms from Paul D. Hanna)

Cf. A227620, A227543, A039924, A005169, A286744, A286745, A003106.

a(n)=polcoeff(sum(m=0, sqrtint(n), (-1)^m*x^(m*(m+1))/prod(k=1, m, 1-x^k,1+x*O(x^n))),n)
for(n=0, 80, print1(a(n), ", "))

A286041 Number of partitions of n with parts differing by at least two and with an even number of parts.

Original entry on oeis.org

1, 0, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 8, 8, 10, 11, 14, 15, 19, 21, 26, 29, 35, 39, 47, 52, 61, 68, 79, 87, 100, 110, 126, 138, 156, 171, 193, 211, 237, 259, 290, 317, 354, 387, 432, 472, 525, 575, 639, 699, 776, 849, 941, 1030, 1139, 1246, 1377, 1505, 1659, 1813, 1996, 2178, 2394, 2610, 2863, 3119, 3415, 3716, 4064, 4416, 4820, 5234, 5705, 6187, 6735, 7297, 7932

Offset: 0

George Beck, May 06 2017

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..5000

Cf. A003114, A286316, A039924.

Table[Length@ Select[IntegerPartitions@n, Min[-Differences@#] > 1  && EvenQ@Length@# &], {n, 80}]

a(n) = A003114(n) - A286316(n).

a(n) ~ phi^(1/2) * exp(2*Pi*sqrt(n/15)) / (4 * 3^(1/4) * 5^(1/2) * n^(3/4)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Mar 05 2020

a(0)=1 prepended by Alois P. Heinz, May 07 2017

A286316 Number of partitions of n with parts differing by at least two and with an odd number of parts.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 9, 11, 13, 15, 17, 20, 22, 25, 28, 32, 35, 40, 44, 50, 56, 63, 70, 80, 89, 101, 113, 128, 143, 162, 181, 204, 228, 256, 285, 320, 355, 396, 439, 489, 540, 599, 660, 730, 803, 886, 972, 1070, 1172, 1287, 1408, 1544, 1686, 1846, 2014, 2202, 2400, 2621, 2854, 3114, 3389, 3693, 4016, 4374, 4753, 5172, 5617, 6107, 6628, 7201, 7810

Offset: 0

George Beck, May 06 2017

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..5000

Cf. A003114, A286041, A039924.

Table[Length@
  Select[IntegerPartitions@n, Min[-Differences@#] > 1 && OddQ@Length@# &], {n, 80}]

a(n) = A003114(n) - A286041(n).

a(n) ~ phi^(1/2) * exp(2*Pi*sqrt(n/15)) / (4 * 3^(1/4) * 5^(1/2) * n^(3/4)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Mar 05 2020

a(0)=0 prepended by Alois P. Heinz, May 07 2017

A350310 Alternating row sums of A066448.

Original entry on oeis.org

1, -1, -2, -2, -1, 0, 2, 4, 6, 7, 8, 8, 6, 4, 0, -6, -11, -18, -26, -32, -38, -44, -46, -46, -44, -37, -26, -12, 8, 32, 58, 90, 124, 158, 194, 228, 259, 286, 306, 316, 316, 304, 276, 232, 170, 88, -12, -132, -272, -431, -606, -794, -994, -1200, -1408, -1614, -1808, -1984, -2138, -2258, -2336

Offset: 0

Seiichi Manyama, Jan 12 2022

sign

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A039924, A066447, A066448, A237979, A350737, A350738.

my(N=66, x='x+O('x^N)); Vec(sum(k=0, sqrtint(N), (-1)^k*x^k^2*prod(j=1, k, (1+x^j)/(1-x^j))))

a(n) = Sum_{k=0..n} (-1)^k * A066448(n,k).

G.f.: Sum_{k>=0} (-1)^k * x^(k^2) * Product_{j=1..k} (1+x^j)/(1-x^j).

A350738 Expansion of Sum_{k>=0} (-1)^k * x^(k^2) * Product_{j=1..k} (1+x^j).

Original entry on oeis.org

1, -1, -1, 0, 1, 1, 1, 1, 0, -1, -1, -1, -2, -1, -1, -1, 1, 1, 1, 2, 2, 2, 2, 2, 1, 0, 0, -1, -2, -2, -3, -3, -3, -3, -3, -3, -1, -1, 0, 1, 1, 3, 4, 4, 4, 5, 5, 5, 5, 3, 3, 3, 1, 0, -1, -3, -4, -4, -6, -7, -7, -8, -8, -8, -7, -7, -6, -5, -4, -2, -1, 1, 3, 5, 6, 8, 9, 10, 12, 13, 13, 12, 13, 12, 11, 11, 9, 7, 5, 3, 0

Offset: 0

Seiichi Manyama, Jan 12 2022

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A039924, A306734, A350310, A350737.

my(N=99, x='x+O('x^N)); Vec(sum(k=0, sqrtint(N), (-1)^k*x^k^2*prod(j=1, k, 1+x^j)))

from math import prod, isqrt
from sympy import Poly
from sympy.abc import x
def A350738(n): return Poly(sum((-1 if k % 2 else 1)*x**(k**2)*prod(1+x**j for j in range(1,k+1)) for k in range(isqrt(n+1)+1))).all_coeffs()[-n-1] # Chai Wah Wu, Jan 14 2022

A350737 Expansion of Sum_{k>=0} (-1)^k * x^(k(3k+1)/2) / Product_{j=1..k} (1-x^j).

Original entry on oeis.org

1, 0, -1, -1, -1, -1, -1, 0, 0, 1, 1, 2, 2, 3, 3, 3, 3, 3, 2, 2, 1, 0, -1, -2, -4, -5, -6, -8, -9, -10, -11, -12, -12, -13, -12, -12, -11, -10, -7, -6, -3, 0, 4, 7, 12, 15, 21, 25, 30, 34, 40, 43, 48, 51, 55, 56, 59, 59, 60, 58, 56, 52, 49, 42, 35, 26, 17, 5, -7, -21, -36, -52, -69, -87, -105, -124, -144, -163

Offset: 0

Seiichi Manyama, Jan 12 2022

sign

Cf. A039924, A237979, A350310, A350738.

my(N=99, x='x+O('x^N)); Vec(sum(k=0, N, (-1)^k*x^(k*(3*k+1)/2)/prod(j=1, k, 1-x^j)))

A227620 Logarithmic derivative of A005169, the number of fountains of n coins.

Original entry on oeis.org

1, 1, 4, 5, 11, 22, 36, 69, 121, 221, 386, 686, 1210, 2122, 3734, 6517, 11408, 19903, 34714, 60485, 105312, 183272, 318758, 554262, 963361, 1674076, 2908426, 5052066, 8774386, 15237482, 26458718, 45939797, 79759442, 138468656, 240382216, 417289619, 724369536, 1257396992

Offset: 1

Paul D. Hanna, Jul 17 2013

nonn

			L.g.f.: L(x) = x + x^2/2 + 4*x^3/3 + 5*x^4/4 + 11*x^5/5 + 22*x^6/6 +...
such L(x) = log(P(x)) - log(Q(x)) where
P(x) = 1 - x^2 - x^3 - x^4 - x^5 + x^8 + x^9 + 2*x^10 + 2*x^11 + 2*x^12 + 2*x^13 + 2*x^14 + x^15 + x^16 - x^18 +...+ A224898(n)*x^n +...
Q(x) = 1 - x - x^2 - x^3 + x^6 + x^7 + 2*x^8 + x^9 + 2*x^10 + x^11 + x^12 - 2*x^15 - x^16 - 3*x^17 - 3*x^18 +...+ A039924(n)*x^n +...
log(P(x)) = -2*x^2/2 - 3*x^3/3 - 6*x^4/4 - 10*x^5/5 - 11*x^6/6 - 21*x^7/7 - 22*x^8/8 - 39*x^9/9 - 42*x^10/10 +...
log(Q(x)) = -x - 3*x^2/2 - 7*x^3/3 - 11*x^4/4 - 21*x^5/5 - 33*x^6/6 - 57*x^7/7 - 91*x^8/8 - 160*x^9/9 - 263*x^10/10 +...

Cf. A227543, A005169, A039924, A224898.

/* As the log of a continued fraction: */
{a(n)=local(A=x, CF=1+x); for(k=0, n, CF=1/(1-x^(n-k+1)*CF+x*O(x^n)); A=log(CF)); n*polcoeff(A, n)}
for(n=1,40,print1(a(n),", "))

/* By the Rogers-Ramanujan continued fraction identity: */
{a(n)=local(A=x, P=1+x, Q=1);
P=sum(m=0, sqrtint(n), (-1)^m*x^(m*(m+1))/prod(k=1, m, 1-x^k));
Q=sum(m=0, sqrtint(n), (-1)^m*x^(m^2)/prod(k=1, m, 1-x^k));
A=log(P/(Q+x*O(x^n))); n*polcoeff(A, n)}
for(n=1,40,print1(a(n),", "))

L.g.f.: log( 1/(1-x/(1-x^2/(1-x^3/(1-x^4/(1-x^5/(1-...)))))) ), the logarithm of a continued fraction.
L.g.f.: log( P(x) / Q(x) ) where
P(x) = Sum_{n>=0} (-1)^n* x^(n*(n+1)) / Product_{k=1..n} (1-x^k),
Q(x) = Sum_{n>=0} (-1)^n* x^(n^2) / Product_{k=1..n} (1-x^k),
due to the Rogers-Ramanujan continued fraction identity.

cp's OEIS Frontend

A003116 Expansion of the reciprocal of the g.f. defining A039924.

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A224898 G.f.: Sum_{n>=0} (-1)^n* x^(n*(n+1)) / Product_{k=1..n} (1-x^k).

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A286041 Number of partitions of n with parts differing by at least two and with an even number of parts.

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A286316 Number of partitions of n with parts differing by at least two and with an odd number of parts.

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A350310 Alternating row sums of A066448.

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A350738 Expansion of Sum_{k>=0} (-1)^k * x^(k^2) * Product_{j=1..k} (1+x^j).

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A350737 Expansion of Sum_{k>=0} (-1)^k * x^(k*(3*k+1)/2) / Product_{j=1..k} (1-x^j).

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A227620 Logarithmic derivative of A005169, the number of fountains of n coins.

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PARI

A350737 Expansion of Sum_{k>=0} (-1)^k * x^(k(3k+1)/2) / Product_{j=1..k} (1-x^j).