A066447 -id:A066447 - cp's OEIS frontend

A376841 Decimal expansion of a constant related to the asymptotics of A066447 and A333374.

7, 1, 5, 7, 8, 7, 4, 1, 7, 8, 6, 1, 4, 3, 5, 2, 4, 8, 8, 0, 2, 0, 5, 0, 1, 6, 4, 9, 9, 8, 9, 1, 0, 1, 6, 0, 6, 4, 8, 2, 6, 7, 9, 7, 5, 9, 3, 5, 4, 9, 3, 7, 3, 6, 1, 9, 5, 7, 5, 8, 6, 2, 7, 2, 5, 2, 3, 3, 7, 2, 3, 7, 1, 3, 7, 9, 3, 2, 6, 7, 7, 9, 3, 1, 5, 5, 3, 5, 7, 1, 4, 2, 1, 6, 4, 3, 3, 3, 7, 8, 6, 9, 0, 6, 6

Offset: 1

Vaclav Kotesovec, Oct 06 2024

			7.1578741786143524880205016499891016064826797593549373619575862725233...

Cf. A066447, A192918, A333374.

RealDigits[E^(2*Sqrt[Log[r]^2 + PolyLog[2, r^2] - PolyLog[2, -r^2]]) /. r -> (-1 - 2/(17 + 3*Sqrt[33])^(1/3) + (17 + 3*Sqrt[33])^(1/3))/3, 10, 105][[1]]

Equals limit_{n->infinity} A066447(n)^(1/sqrt(n)).
Equals limit_{n->infinity} A333374(n)^(1/sqrt(n)).
Equals exp(2*sqrt(log(r)^2 - polylog(2, -r^2) + polylog(2, r^2))), where r = A192918 = 0.54368901269207636157... is the real root of the equation r^2*(1+r) = 1-r.

A333374 G.f.: Sum_{k>=1} (x^(k(k+1)) Product_{j=1..k} (1 + x^j)/(1 - x^j)).

Original entry on oeis.org

1, 0, 1, 2, 2, 2, 3, 4, 6, 8, 10, 12, 15, 18, 22, 28, 34, 42, 52, 62, 75, 90, 106, 126, 150, 176, 208, 246, 288, 338, 397, 462, 538, 626, 724, 838, 968, 1114, 1282, 1474, 1690, 1936, 2217, 2532, 2890, 3296, 3750, 4264, 4844, 5492, 6222, 7042, 7958, 8986, 10138

Offset: 0

Vaclav Kotesovec, Mar 17 2020

nonn

The g.f. Sum_{k >= 1} x^(k*(k+1)) * Product_{j = 1..k} (1 + x^j)/(1 - x^j) = Sum_{k >= 1} x^(k*(k+1)) * Product_{j = 1..k} (1 + x^j)/(1 - x^j + 2*x^j) == Sum_{k >= 1} x^(k*(k+1)) (mod 2). It follows that a(n) is odd iff n = k*(k + 1) for some nonnegative integer k. - Peter Bala, Jan 04 2025

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A001935, A001936, A003106, A053261, A066447, A003114, A306734, A333179.

Cf. A192918, A376841.

nmax = 100; CoefficientList[Series[Sum[x^(n*(n+1))*Product[(1+x^k)/(1-x^k), {k, 1, n}], {n, 0, Sqrt[nmax]}], {x, 0, nmax}], x]
nmax = 100; p = 1; s = 1; Do[p = Normal[Series[p*(1 + x^k)/(1 - x^k)*x^(2*k), {x, 0, nmax}]]; s += p;, {k, 1, Sqrt[nmax]}]; Take[CoefficientList[s, x], nmax + 1]

Limit_{n->infinity} A066447(n) / a(n) = A058265 = (1 + (19+3*sqrt(33))^(1/3) + (19-3*sqrt(33))^(1/3))/3 = 1.839286755214... (the tribonacci constant).
Compare with: A306734(n) / A333179(n) -> A060006 (the plastic constant) and A003114(n) / A003106(n) -> A001622 (golden ratio).
a(n) ~ c * d^sqrt(n) / n^(3/4), where d = A376841 = 7.1578741786143524880205... = exp(2*sqrt(log(r)^2 - polylog(2, -r^2) + polylog(2, r^2))) and c = 0.10511708841962944170826735560432... = (log(r)^2 - polylog(2, -r^2) + polylog(2, r^2))^(1/4) * sqrt(1/24 - sinh(arcsinh(sqrt(11)/4)/3) / (12*sqrt(11))) / sqrt(Pi), where r = A192918 = 0.54368901269207636157... is the real root of the equation r^2*(1+r) = 1-r. - Vaclav Kotesovec, Mar 17 2020, updated Oct 10 2024

A066448 Triangle T(n,k) giving number of basis partitions of n with a Durfee square of order k (n >= 0, 0 <= k <= n).

Original entry on oeis.org

1, 0, 1, 0, 2, 0, 0, 2, 0, 0, 0, 2, 1, 0, 0, 0, 2, 2, 0, 0, 0, 0, 2, 4, 0, 0, 0, 0, 0, 2, 6, 0, 0, 0, 0, 0, 0, 2, 8, 0, 0, 0, 0, 0, 0, 0, 2, 10, 1, 0, 0, 0, 0, 0, 0, 0, 2, 12, 2, 0, 0, 0, 0, 0, 0, 0, 0, 2, 14, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 16, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

N. J. A. Sloane, Dec 29 2001

			Triangle begins:
  1;
  0, 1;
  0, 2, 0;
  0, 2, 0, 0;
  0, 2, 1, 0, 0;
  0, 2, 2, 0, 0, 0;
  0, 2, 4, 0, 0, 0, 0;
  0, 2, 6, 0, 0, 0, 0, 0;
  0, 2, 8, 0, 0, 0, 0, 0, 0;
  ...

Seiichi Manyama, Rows n = 0..139, flattened
J. M. Nolan, C. D. Savage and H. S. Wilf, Basis partitions, Discrete Math. 179 (1998), 277-283.

Row sums give A066447.

T := proc(n,d); option remember; if n=0 and d=0 then RETURN(1) elif n<=0 or d<=0 then RETURN(0) else RETURN(T(n-d,d)+T(n-2*d+1,d-1)+T(n-3*d+1,d-1)) fi:

T(n,k)=if(k<0||k>n,0,if(k==0,n==0,T(n-k,k)+T(n-2*k+1,k-1)+T(n-3*k+1,k-1))) /* Michael Somos, Mar 10 2004 */

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).

A376854 G.f.: Sum_{k>=0} x^(k^2) * Product_{j=1..k} ((1 + x^j)/(1 - x^j))^2.

Original entry on oeis.org

1, 1, 4, 8, 13, 20, 32, 52, 84, 133, 204, 304, 444, 636, 900, 1264, 1761, 2440, 3364, 4608, 6276, 8496, 11424, 15268, 20284, 26789, 35196, 46016, 59884, 77612, 100204, 128900, 165260, 211200, 269072, 341792, 432917, 546788, 688728, 865200, 1084048, 1354816, 1689048

Offset: 0

Vaclav Kotesovec, Oct 06 2024

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A066447, A000041, A216222, A376852, A376853.

nmax = 60; CoefficientList[Series[Sum[x^(k^2) * Product[(1+x^j)/(1-x^j), {j, 1, k}]^2, {k, 0, Sqrt[nmax]}], {x, 0, nmax}], x]

a(n) ~ (1 + sqrt(2)) * exp(Pi*sqrt(n)) / (2^(9/2) * n).

A001130 Number of graphical basis partitions of 2n.

Original entry on oeis.org

1, 1, 3, 4, 6, 11, 16, 23, 36, 52, 71, 103, 141, 197, 272, 366, 482, 657, 863, 1140, 1489, 1951, 2511, 3241, 4155, 5317, 6782, 8574, 10786, 13645, 17111, 21313, 26631, 33020, 41005, 50640, 62373, 76510, 94089, 114991, 140376, 170970, 207837, 251552, 305342, 368474, 444360, 534692, 642593, 770278

Offset: 1

Pranav Kumar Tiwari (pktiwari(AT)eos.ncsu.edu)

nonn

A partition of an even integer is graphical if it is the degree sequence of a simple graph.

Nolan, Jennifer M.; Sivaraman, Vijay; Savage, Carla D.; and Tiwari, Pranav K., Graphical basis partitions, Graphs Combin. 14 (1998), no. 3, 241-261. Math. Rev. 99j:05014. See http://www4.ncsu.edu/~savage/papers.html for postscript file.

Ray Chandler, Table of n, a(n) for n = 1..100 [from the Nolan et al. paper]
Index entries for sequences related to graphical partitions

Cf. A000041, A000569, A066447.

Seven more terms (all that are presently known, apparently) added from the Nolan et al. paper by N. J. A. Sloane, Jun 01 2012

Extended b-file from Nolan et al. paper and adjusted description to even n by Ray Chandler, Sep 17 2015

A350636 a(n) is the number of partitions of n in which no part is divisible by 3 minus the number of basis partitions of n.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 1, 1, 3, 3, 6, 7, 10, 12, 17, 20, 28, 34, 45, 55, 72, 87, 111, 133, 167, 200, 247, 295, 362, 431, 523, 621, 749, 885, 1059, 1247, 1482, 1739, 2055, 2402, 2826, 3293, 3855, 4479, 5225, 6051, 7032, 8123, 9406, 10837, 12509, 14372, 16541, 18960, 21756, 24880, 28477

Offset: 0

Michel Marcus, Jan 09 2022

nonn

Andrews conjectures that a(n)>0 for n>3.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
George E. Andrews, Partition Identities for Two-Color Partitions, Hardy-Ramanujan Journal, Hardy-Ramanujan Society, 2021, Special Commemorative volume in honour of Srinivasa Ramanujan, 2021, 44, pp.74-80. hal-03498190. See p. 79.

Cf. A000726, A066447.

a(n) = A000726(n) - A066447(n).

cp's OEIS Frontend

A376841 Decimal expansion of a constant related to the asymptotics of A066447 and A333374.

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A333374 G.f.: Sum_{k>=1} (x^(k*(k+1)) * Product_{j=1..k} (1 + x^j)/(1 - x^j)).

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A066448 Triangle T(n,k) giving number of basis partitions of n with a Durfee square of order k (n >= 0, 0 <= k <= n).

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Maple

PARI

A350310 Alternating row sums of A066448.

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PARI

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A376854 G.f.: Sum_{k>=0} x^(k^2) * Product_{j=1..k} ((1 + x^j)/(1 - x^j))^2.

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A001130 Number of graphical basis partitions of 2n.

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A350636 a(n) is the number of partitions of n in which no part is divisible by 3 minus the number of basis partitions of n.

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A333374 G.f.: Sum_{k>=1} (x^(k(k+1)) Product_{j=1..k} (1 + x^j)/(1 - x^j)).