A266232 -id:A266232 - cp's OEIS frontend

A281425 a(n) = [q^n] (1 - q)^n / Product_{j=1..n} (1 - q^j).

1, 0, 1, -1, 2, -4, 9, -21, 49, -112, 249, -539, 1143, -2396, 5013, -10550, 22420, -48086, 103703, -223806, 481388, -1029507, 2187944, -4625058, 9742223, -20490753, 43111808, -90840465, 191773014, -405523635, 858378825, -1817304609, 3845492204, -8129023694, 17162802918, -36191083386

Offset: 0

Ilya Gutkovskiy, Oct 05 2017

sign

a(n) is n-th term of the Euler transform of -n + 1, 1, 1, 1, ...

Inverse zero-based binomial transform of A000041. The version for strict partitions is A380412, or A293467 up to sign. - Gus Wiseman, Feb 06 2025

Vaclav Kotesovec, Table of n, a(n) for n = 0..3000
A. M. Odlyzko, Differences of the partition function, Acta Arithmetica 49.3 (1988): 237-254.
Dennis Stanton and Doron Zeilberger, The Odlyzko conjecture and O’Hara’s unimodality proof, Proceedings of the American Mathematical Society 107.1 (1989): 39-42.

Cf. A128566, A292463, A292541, A292613.
Cf. A000041, A002865, A053445, A275638, A275639, A275640, A275641, A275642, A275643, A275644.
Cf. A218481, A294466, A095051.
Cf. A266232, A294467, A293467, A294468.
Row n=0 of A175804 (row n=1 is A320590 except first term).
Cf. A000009, A008284, A087897, A116608, A129519, A225486, A320591, A377056, A378622.

b:= proc(n, k) option remember; `if`(k=0,
      combinat[numbpart](n), b(n, k-1)-b(n-1, k-1))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..35);  # Alois P. Heinz, Dec 21 2024

Table[SeriesCoefficient[(1 - q)^n / Product[(1 - q^j), {j, 1, n}], {q, 0, n}], {n, 0, 35}]
Table[SeriesCoefficient[(1 - q)^n QPochhammer[q^(1 + n), q]/QPochhammer[q, q], {q, 0, n}], {n, 0, 35}]
Table[SeriesCoefficient[1/QFactorial[n, q], {q, 0, n}], {n, 0, 35}]
Table[Differences[PartitionsP[Range[0, n]], n], {n, 0, 35}] // Flatten
Table[Sum[(-1)^j*Binomial[n, j]*PartitionsP[n-j], {j, 0, n}], {n, 0, 30}] (* Vaclav Kotesovec, Oct 06 2017 *)

a(n) = [q^n] 1/((1 + q)*(1 + q + q^2)*...*(1 + q + ... + q^(n-1))).
a(n) = Sum_{j=0..n} (-1)^j * binomial(n, j) * A000041(n-j). - Vaclav Kotesovec, Oct 06 2017
a(n) ~ (-1)^n * 2^(n - 3/2) * exp(Pi*sqrt(n/12) + Pi^2/96) / (sqrt(3)*n). - Vaclav Kotesovec, May 07 2018

A218481 Binomial transform of the partition numbers (A000041).

Original entry on oeis.org

1, 2, 5, 13, 34, 88, 225, 569, 1425, 3538, 8717, 21331, 51879, 125474, 301929, 723144, 1724532, 4096210, 9693455, 22859524, 53733252, 125919189, 294232580, 685661202, 1593719407, 3695348909, 8548564856, 19732115915, 45450793102, 104481137953, 239718272765

Offset: 0

Paul D. Hanna, Oct 29 2012

Partial sums of A218482.
From Vaclav Kotesovec, Nov 02 2023: (Start)
Let 0 < p < 1, r > 0, v > 0, f(n) = v*exp(r*n^p)/n^b, then
Sum_{k=0..n} binomial(n,k) * f(k) ~ f(n/2) * 2^n * exp(g(n)), where
g(n) = p^2 * r^2 * n^p / (2^(1+2*p)*n^(1-p) + p*r*(1-p)*2^(1+p)).
Special cases:
p < 1/2, g(n) = 0
p = 1/2, g(n) = r^2/16
p = 2/3, g(n) = r^2 * n^(1/3) / (9 * 2^(1/3)) - r^3/81
p = 3/4, g(n) = 9*r^2*sqrt(n)/(64*sqrt(2)) - 27*r^3*n^(1/4)/(2048*2^(1/4)) + 81*r^4/65536
p = 3/5, g(n) = 9*r^2*n^(1/5)/(100*2^(1/5))
p = 4/5, g(n) = 2^(7/5)*r^2*n^(3/5)/25 - 4*2^(3/5)*r^3*n^(2/5)/625 + 8*2^(4/5)*r^4*n^(1/5)/15625 - 32*r^5/390625
(End)

			G.f.: A(x) = 1 + 2*x + 5*x^2 + 13*x^3 + 34*x^4 + 88*x^5 + 225*x^6 + 569*x^7 +...
The g.f. equals the product:
A(x) = 1/((1-x)-x) * (1-x)^2/((1-x)^2-x^2) * (1-x)^3/((1-x)^3-x^3) * (1-x)^4/((1-x)^4-x^4) * (1-x)^5/((1-x)^5-x^5) * (1-x)^6/((1-x)^6-x^6) * (1-x)^7/((1-x)^7-x^7) *...
and also equals the series:
A(x) = 1/(1-x) * (1  +  x*(1-x)/((1-x)-x)^2  +  x^4*(1-x)^2/(((1-x)-x)*((1-x)^2-x^2))^2  +  x^9*(1-x)^3/(((1-x)-x)*((1-x)^2-x^2)*((1-x)^3-x^3))^2  +  x^16*(1-x)^4/(((1-x)-x)*((1-x)^2-x^2)*((1-x)^3-x^3)*((1-x)^4-x^4))^2 +...).
The terms begin:
a(0) = 1*1,
a(1) = 1*1 + 1*1 = 2;
a(2) = 1*1 + 2*1 + 1*2 = 5;
a(3) = 1*1 + 3*1 + 3*2 + 1*3 = 13;
a(4) = 1*1 + 4*1 + 6*2 + 4*3 + 1*5 = 34; ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A218482, A222115, A000041, A000203, A266497.
Cf. A294466, A281425, A095051.
Cf. A266232, A294467, A293467, A294468, A294500.

Table[Sum[Binomial[n,k]*PartitionsP[k],{k,0,n}],{n,0,30}] (* Vaclav Kotesovec, Jun 25 2015 *)
nmax = 30; CoefficientList[Series[Sum[PartitionsP[k] * x^k / (1-x)^(k+1), {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 31 2022 *)

{a(n)=sum(k=0,n,binomial(n,k)*numbpart(k))}
for(n=0,40,print1(a(n),", "))

{a(n)=local(X=x+x*O(x^n));polcoeff(1/(1-X)*prod(k=1,n,(1-x)^k/((1-x)^k-X^k)),n)}

{a(n)=local(X=x+x*O(x^n));polcoeff(1/(1-X)*sum(m=0,n,x^m*(1-x)^(m*(m-1)/2)/prod(k=1,m,((1-x)^k - X^k))),n)}

{a(n)=local(X=x+x*O(x^n));polcoeff(1/(1-X)*sum(m=0,n,x^(m^2)*(1-X)^m/prod(k=1,m,((1-x)^k - x^k)^2)),n)}

{a(n)=local(X=x+x*O(x^n));polcoeff(1/(1-X)*exp(sum(m=1,n+1,x^m/((1-x)^m-X^m)/m)),n)}

{a(n)=local(X=x+x*O(x^n));polcoeff(1/(1-X)*exp(sum(m=1,n+1,sigma(m)*x^m/(1-X)^m/m)),n)}

{a(n)=local(X=x+x*O(x^n));polcoeff(1/(1-X)*prod(k=1,n,(1 + x^k/(1-X)^k)^valuation(2*k,2)),n)}

G.f.: 1/(1-x)*Product_{n>=1} (1-x)^n / ((1-x)^n - x^n).
G.f.: 1/(1-x)*Sum_{n>=0} x^n * (1-x)^(n*(n-1)/2) / Product_{k=1..n} ((1-x)^k - x^k).
G.f.: 1/(1-x)*Sum_{n>=0} x^(n^2) * (1-x)^n / Product_{k=1..n} ((1-x)^k - x^k)^2.
G.f.: 1/(1-x)*exp( Sum_{n>=1} x^n/((1-x)^n - x^n) / n ).
G.f.: 1/(1-x)*exp( Sum_{n>=1} sigma(n) * x^n/(1-x)^n / n ), where sigma(n) is the sum of divisors of n (A000203).
G.f.: 1/(1-x)*Product_{n>=1} (1 + x^n/(1-x)^n)^A001511(n), where 2^A001511(n) is the highest power of 2 that divides 2*n.
Logarithmic derivative yields A222115.
a(n) ~ exp(Pi*sqrt(n/3) + Pi^2/24) * 2^(n-1) / (n*sqrt(3)). - Vaclav Kotesovec, Jun 25 2015

A293467 a(n) = Sum_{k=0..n} (-1)^k * binomial(n, k) * q(k), where q(k) is A000009 (partitions into distinct parts).

Original entry on oeis.org

1, 0, 0, -1, -3, -7, -14, -25, -41, -64, -100, -165, -294, -550, -1023, -1795, -2823, -3658, -2882, 2873, 20435, 62185, 148863, 314008, 613957, 1155794, 2175823, 4244026, 8753538, 19006490, 42471787, 95234575, 210395407, 453413866, 949508390, 1931939460

Offset: 0

Vaclav Kotesovec, Oct 09 2017

sign

Multiply by (-1)^n to get A380412, which is the first term of the n-th differences of the strict partition numbers, or column n=0 of A378622. - Gus Wiseman, Feb 04 2025

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, Graph: a(n)/2^n (40000 terms)

Cf. A025147, A266232, A294467, A294468.
Cf. A218481, A294466, A095051.
The non-strict version is the absolute value of A281425; see A175804, A320590.
Up to sign, same as A380412. See A320591, A377285, A378970, A378971.
A000009 counts strict integer partitions, differences A087897.
Cf. A000041, A002865, A007442, A008284, A116608.

Table[Sum[(-1)^k * Binomial[n, k] * PartitionsQ[k], {k, 0, n}], {n, 0, 50}]

A129519 First differences of the binomial transform of the distinct partition numbers (A000009).

Original entry on oeis.org

1, 1, 2, 5, 12, 28, 65, 151, 350, 807, 1850, 4221, 9597, 21760, 49215, 111032, 249856, 560835, 1255854, 2805969, 6256784, 13925698, 30941050, 68634679, 152009239, 336152787, 742276931, 1636747349, 3604206106, 7926412320, 17410413153

Offset: 0

Paul D. Hanna, Apr 18 2007

nonn

			Product formula is illustrated by:
A(x) = [1 + x + x^2 + x^3 + x^4 + x^5 +...]*
[1 + x^2 + 2x^3 + 3x^4 + 4x^5 + 5x^6 +...]*
[1 + x^3 + 3x^4 + 6x^5 + 10x^6 + 15x^7 +...]*
[1 + x^4 + 4x^5 + 10x^6 + 20x^7 + 35x^8 +...]*
[1 + x^5 + 5x^6 + 15x^7 + 35x^8 + 70x^9 +...]*...*
[1 + Sum_{k>=n+1} C(k-1,n)*x^k ]*...

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A000009, A218482, A266232, A307501.

Flatten[{1, Differences[Table[Sum[Binomial[n, k]*PartitionsQ[k], {k, 0, n}], {n, 0, 40}]]}] (* Vaclav Kotesovec, Oct 30 2017 *)

{a(n)=polcoeff(prod(k=0,n,1+sum(i=k+1,n,binomial(i-1,k)*x^i +x*O(x^n))),n)}

G.f.: A(x) = Product_{n>=1} [1 + x^n/(1-x)^n].
a(n) = A266232(n) - A266232(n-1), for n>0. - Vaclav Kotesovec, Oct 30 2017
a(n) ~ exp(Pi*sqrt(n/6) + Pi^2/48) * 2^(n - 9/4) / (3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Oct 30 2017

A294500 Binomial transform of the number of planar partitions (A000219).

Original entry on oeis.org

1, 2, 6, 19, 60, 185, 559, 1662, 4875, 14134, 40564, 115370, 325465, 911355, 2534595, 7004827, 19246626, 52596377, 143006632, 386984573, 1042537831, 2796803110, 7473161196, 19893461042, 52767059608, 139488323734, 367540167625, 965445514862, 2528516552660

Offset: 0

Vaclav Kotesovec, Nov 01 2017

nonn

Let 0 < p < 1, r > 0, v > 0, f(n) = v*exp(r*n^p)/n^b, then
Sum_{k=0..n} binomial(n,k) * f(k) ~ f(n/2) * 2^n * exp(g(n)), where
g(n) = p^2 * r^2 * n^p / (2^(1+2*p)*n^(1-p) + p*r*(1-p)*2^(1+p)).
Special cases:
p < 1/2, g(n) = 0
p = 1/2, g(n) = r^2/16
p = 2/3, g(n) = r^2 * n^(1/3) / (9 * 2^(1/3)) - r^3/81
p = 3/4, g(n) = 9*r^2*sqrt(n)/(64*sqrt(2)) - 27*r^3*n^(1/4)/(2048*2^(1/4)) + 81*r^4/65536
p = 3/5, g(n) = 9*r^2*n^(1/5)/(100*2^(1/5))
p = 4/5, g(n) = 2^(7/5)*r^2*n^(3/5)/25 - 4*2^(3/5)*r^3*n^(2/5)/625 + 8*2^(4/5)*r^4*n^(1/5)/15625 - 32*r^5/390625

Vaclav Kotesovec, Table of n, a(n) for n = 0..2930

Cf. A218481, A266232, A294501, A294502, A294504.

nmax = 40; s = CoefficientList[Series[Product[1/(1-x^k)^k, {k, 1, nmax}], {x, 0, nmax}], x]; Table[Sum[Binomial[n, k] * s[[k+1]], {k, 0, n}], {n, 0, nmax}]

a(n) = Sum_{k=0..n} binomial(n,k) * A000219(k).
a(n) ~ exp(1/12 + 3 * Zeta(3)^(1/3) * n^(2/3) / 2^(4/3) + Zeta(3)^(2/3) * n^(1/3) / 2^(5/3) - Zeta(3)/12) * 2^(n + 7/18) * Zeta(3)^(7/36) / (A * sqrt(3*Pi) * n^(25/36)), where A is the Glaisher-Kinkelin constant A074962.
G.f.: (1/(1 - x))*exp(Sum_{k>=1} sigma_2(k)*x^k/(k*(1 - x)^k)). - Ilya Gutkovskiy, Aug 20 2018

A294466 Binomial transform of A053529.

Original entry on oeis.org

1, 2, 7, 34, 221, 1666, 15187, 153602, 1770169, 22379266, 312164831, 4685997922, 76668261397, 1335425319554, 24921410400811, 493075754663746, 10358312736025457, 228862423291312642, 5335861084579488439, 130235118120543955106, 3333808742649699747661

Offset: 0

Vaclav Kotesovec, Oct 31 2017

nonn

G. C. Greubel, Table of n, a(n) for n = 0..441

Cf. A218481, A281425, A095051.

Cf. A266232, A294467, A293467, A294468.

Table[Sum[Binomial[n, k]*k!*PartitionsP[k], {k, 0, n}], {n, 0, 20}]
nmax = 20; CoefficientList[Series[Exp[x] * x^(1/24)/DedekindEta[Log[x]/(2*Pi*I)], {x, 0, nmax}], x] * Range[0, nmax]!

x='x+O('x^50); Vec(serlaplace(exp(x)/eta(x))) \\ G. C. Greubel, Oct 15 2018

E.g.f.: exp(x)/eta(x), where eta(x) is the Dedekind eta function.
a(n) ~ exp(1) * n! * A000041(n).
a(n) ~ sqrt(2*Pi) * exp(Pi*sqrt(2*n/3) - n + 1) * n^(n - 1/2) / (4*sqrt(3)).
E.g.f.: exp(x + Sum_{k>=1} sigma(k)*x^k/k). - Ilya Gutkovskiy, Oct 15 2018

A095051 E.g.f.: exp(-x)/eta(x), where eta(x) is the Dedekind eta function.

Original entry on oeis.org

1, 0, 3, 8, 69, 384, 4375, 34152, 464457, 5051456, 75865131, 1032865800, 18108977293, 286975230528, 5639956035519, 105513165321704, 2269311347406225, 48066460265622912, 1146324511845384787, 26924271371612501256, 701472699537610875861, 18214089447110112972800, 512194770431254272442983

Offset: 0

Benoit Cloitre, Jun 19 2004

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..440
N. J. A. Sloane, Transforms

Cf. A218481, A294466, A281425.

Cf. A266232, A294467, A293467, A294468.

Table[Sum[(-1)^(n-k) * Binomial[n, k] * k! * PartitionsP[k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 31 2017 *)
nmax = 20; CoefficientList[Series[Exp[-x] * x^(1/24)/DedekindEta[Log[x]/(2*Pi*I)], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Oct 31 2017 *)

PARI
```
a(n)=polcoeff(1/eta(x)/exp(x),n)*n!
```

Inverse binomial transform of A053529. - Vladeta Jovovic, Jun 21 2004
From Vaclav Kotesovec, Oct 31 2017: (Start)
a(n) ~ exp(-1) * n! * A000041(n).
a(n) ~ sqrt(2*Pi) * exp(Pi*sqrt(2*n/3) - n - 1) * n^(n - 1/2) / (4*sqrt(3)). (End)
E.g.f.: exp(Sum_{k>=2} sigma(k)*x^k/k). - Ilya Gutkovskiy, Oct 15 2018

More terms from Michel Marcus, Oct 31 2017

A294467 Binomial transform of A088311.

Original entry on oeis.org

1, 2, 5, 22, 113, 746, 6037, 55070, 548417, 6281938, 79935941, 1087584422, 16109401585, 255667890362, 4358283982613, 79893373511086, 1542859916102657, 31322024816838050, 676027617881188357, 15287136167625123638, 362322855217463741681

Offset: 0

Vaclav Kotesovec, Oct 31 2017

nonn

G. C. Greubel, Table of n, a(n) for n = 0..443

Cf. A266232, A294467, A294468.

Cf. A218481, A294466, A281425, A095051.

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(x)*(&*[1 + x^k: k in [1..50]]))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Oct 15 2018

Table[Sum[Binomial[n, k]*k!*PartitionsQ[k], {k, 0, n}], {n, 0, 20}]

x='x+O('x^30); Vec(serlaplace(exp(x)*eta(x^2)/eta(x))) \\ G. C. Greubel, Oct 15 2018

a(n) ~ exp(1) * n! * A000009(n).
a(n) ~ sqrt(2*Pi) * exp(Pi*sqrt(n/3) - n + 1) * n^(n - 1/4) / (4*3^(1/4)).
E.g.f.: exp(x) * Product_{k>=1} (1 + x^k). - Ilya Gutkovskiy, Oct 15 2018

A294468 Inverse binomial transform of A088311.

Original entry on oeis.org

1, 0, 1, 8, 9, 224, 1225, 11304, 103537, 1431296, 15642801, 206721800, 3295533241, 47467875168, 859354139449, 15596241280424, 283240963555425, 5859309797252864, 129874369387025377, 2752905169704533256, 67640333903657850601

Offset: 0

Vaclav Kotesovec, Oct 31 2017

nonn

G. C. Greubel, Table of n, a(n) for n = 0..443

Cf. A266232, A294467, A293467.

Cf. A218481, A294466, A281425, A095051.

Table[Sum[(-1)^(n-k)*Binomial[n, k]*k!*PartitionsQ[k], {k, 0, n}], {n, 0, 20}]
max = 20; t = Table[k!*PartitionsQ[k], {k, 0, max}]; Table[Differences[t, n], {n, 0, max}][[All, 1]] (* Jean-François Alcover, Nov 02 2017 *)

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * A088311(k).
a(n) ~ exp(-1) * n! * A000009(n).
a(n) ~ sqrt(2*Pi) * exp(Pi*sqrt(n/3) - n - 1) * n^(n - 1/4) / (4*3^(1/4)).
E.g.f.: exp(-x) * Product_{k>=1} (1 + x^k). - Ilya Gutkovskiy, Oct 15 2018

A266497 Binomial transform of A015128.

Original entry on oeis.org

1, 3, 9, 27, 79, 225, 627, 1717, 4633, 12341, 32501, 84737, 218959, 561263, 1428287, 3610671, 9072367, 22668285, 56345835, 139382713, 343242533, 841713531, 2055944117, 5003148987, 12132552115, 29323810757, 70651867863, 169719163521, 406541986857, 971192810019

Offset: 0

Vaclav Kotesovec, Dec 30 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..3000

Cf. A015128, A218481, A266232, A294499.

A015128[n_]:=Sum[PartitionsP[n-k]*PartitionsQ[k], {k, 0, n}];
Table[Sum[Binomial[n, k]*A015128[k], {k, 0, n}], {n, 0, 30}]

a(n) ~ 2^(n-2) * exp(Pi*sqrt(n/2) + Pi^2/16) / n.

a(n) = [x^n] (1 + x)^n/theta_4(x), where theta_4() is the Jacobi theta function. - Ilya Gutkovskiy, Apr 20 2018

cp's OEIS Frontend

A281425 a(n) = [q^n] (1 - q)^n / Product_{j=1..n} (1 - q^j).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A218481 Binomial transform of the partition numbers (A000041).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

PARI

PARI

PARI

PARI

Formula

A293467 a(n) = Sum_{k=0..n} (-1)^k * binomial(n, k) * q(k), where q(k) is A000009 (partitions into distinct parts).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A129519 First differences of the binomial transform of the distinct partition numbers (A000009).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A294500 Binomial transform of the number of planar partitions (A000219).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A294466 Binomial transform of A053529.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A095051 E.g.f.: exp(-x)/eta(x), where eta(x) is the Dedekind eta function.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula