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.

Showing 1-7 of 7 results.

A015128 Number of overpartitions of n: an overpartition of n is an ordered sequence of nonincreasing integers that sum to n, where the first occurrence of each integer may be overlined.

Original entry on oeis.org

1, 2, 4, 8, 14, 24, 40, 64, 100, 154, 232, 344, 504, 728, 1040, 1472, 2062, 2864, 3948, 5400, 7336, 9904, 13288, 17728, 23528, 31066, 40824, 53408, 69568, 90248, 116624, 150144, 192612, 246256, 313808, 398640, 504886, 637592, 802936, 1008448
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

The over-partition function.
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Also the number of jagged partitions of n.
According to Ramanujan (1913) a(n) is close to (cosh(x)-sinh(x)/x)/(4*n) where x=Pi*sqrt(n). - Michael Somos, Mar 17 2003
Number of partitions of 2n with all odd parts occurring with even multiplicities. There is no restriction on the even parts. Cf. A006950, A046682. - Mamuka Jibladze, Sep 05 2003
Number of partitions of n where there are two kinds of odd parts. - Joerg Arndt, Jul 30 2011. Or, in Gosper's words, partitions into red integers and blue odd integers. - N. J. A. Sloane, Jul 04 2016.
Coincides with the sequence of numbers of nilpotent conjugacy classes in the Lie algebras sp(n), n=0,1,2,3,... (the case n=0 being degenerate). A006950, this sequence and A000041 together cover the nilpotent conjugacy classes in the classical A,B,C,D series of Lie algebras. - Alexander Elashvili, Sep 08 2003
Also, number of 01-partitions of n. A 01-partition of n is a weakly decreasing sequence of m nonnegative integers n(i) such that sum(n(i))=n, n(m)>0, n(j)>=n(j+1)-1 and n(j)>=n(j+2). They are special cases of jagged partitions.
a(8n+7) is divisible by 64 (from Fortin/Jacob/Mathieu paper).
Smallest sequence of even numbers (except a(0)) which is the Euler transform of a sequence of positive integers. - Franklin T. Adams-Watters, Oct 16 2006
Convolution of A000041 and A000009. - Vladeta Jovovic, Nov 26 2002
Equals A022567 convolved with A035363. - Gary W. Adamson, Jun 09 2009
Equals the infinite product [1,2,2,2,...] * [1,0,2,0,2,0,2,...] * [1,0,0,2,0,0,2,0,0,2,...] * ... . - Gary W. Adamson, Jul 05 2009
Equals A182818 convolved with A010815. - Gary W. Adamson, Jul 20 2012
Partial sums of A211971. - Omar E. Pol, Jan 09 2014
Also 1 together with the row sums of A235790. - Omar E. Pol, Jan 19 2014
Antidiagonal sums of A284592. - Peter Bala, Mar 30 2017
The overlining method is equivalent to enumerating the k-subsets of the distinct parts of the i-th partition. - Richard Joseph Boland, Sep 02 2021

Examples

			G.f. = 1 + 2*q + 4*q^2 + 8*q^3 + 14*q^4 + 24*q^5 + 40*q^6 + 64*q^7 + 100*q^8 + ...
For n = 4 the 14 overpartitions of 4 are [4], [4'], [2, 2], [2', 2], [3, 1], [3', 1], [3, 1'], [3', 1'], [2, 1, 1], [2', 1, 1], [2, 1', 1], [2', 1', 1], [1, 1, 1, 1], [1', 1, 1, 1]. - _Omar E. Pol_, Jan 19 2014

References

  • J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 103.
  • R. W. Gosper, Experiments and discoveries in q-trigonometry, in Symbolic Computation, Number Theory, Special Functions, Physics and Combinatorics. Editors: F. G. Garvan and M. E. H. Ismail. Kluwer, Dordrecht, Netherlands, 2001, pp. 79-105. See the function g(q).
  • James R. Newman, The World of Mathematics, Simon and Schuster, 1956, Vol. I p. 372.

Links

Crossrefs

Cf. A022567, A035363, A002448, A106507, A156616, A261386, A265835, A014968, A284592, A303662.
See A004402 for a version with signs.
Column k=2 of A321884.
Cf. A002513.

Programs

  • Julia
    # JacobiTheta4 is defined in A002448.
A015128List(len) = JacobiTheta4(len, -1)
A015128List(40) |> println # Peter Luschny, Mar 12 2018
  • Maple
    mul((1+x^n)/(1-x^n),n=1..256): seq(coeff(series(%,x,n+1),x,n), n=0..40);
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1) +2*add(b(n-i*j, i-1), j=1..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..40);  # Alois P. Heinz, Feb 10 2014
a_list := proc(len) series(1/JacobiTheta4(0,x),x,len+1); seq(coeff(%,x,j),j=0..len) end: a_list(39); # Peter Luschny, Mar 14 2017
  • Mathematica
    max = 39; f[x_] := Exp[Sum[(DivisorSigma[1, 2*n] - DivisorSigma[1, n])*(x^n/n), {n, 1, max}]]; CoefficientList[ Series[f[x], {x, 0, max}], x] (* Jean-François Alcover, Jun 11 2012, after Joerg Arndt *)
a[ n_] := SeriesCoefficient[ QHypergeometricPFQ[ {-1}, {}, x, x], {x, 0, n}]; (* Michael Somos, Mar 11 2014 *)
QP = QPochhammer; s = QP[q^2]/QP[q]^2 + O[q]^40; CoefficientList[s + O[q]^100, q] (* Jean-François Alcover, Nov 25 2015, after Michael Somos *)
Table[Sum[PartitionsP[n-k]*PartitionsQ[k], {k, 0, n}], {n, 0, 50}] (* Vaclav Kotesovec, Nov 28 2015 *)
(QPochhammer[-x, x]/QPochhammer[x, x] + O[x]^50)[[3]] (* Vladimir Reshetnikov, Nov 12 2016 *)
nmax = 100; p = ConstantArray[0, nmax+1]; p[[1]] = 1; Do[p[[n+1]] = 0; k = 1; While[n + 1 - k^2 > 0, p[[n+1]] += (-1)^(k+1)*p[[n + 1 - k^2]]; k++;]; p[[n+1]] = 2*p[[n+1]];, {n, 1, nmax}]; p (* Vaclav Kotesovec, Apr 11 2017 *)
a[ n_] := SeriesCoefficient[ 1 / EllipticTheta[ 4, 0, x], {x, 0, n}]; (* Michael Somos, Nov 15 2018 *)
a[n_] := Sum[2^Length[Union[IntegerPartitions[n][[i]]]], {i, 1, PartitionsP[n]}]; (* Richard Joseph Boland, Sep 02 2021 *)
n = 39; CoefficientList[Product[(1 + x^k)/(1 - x^k), {k, 1, n}] + O[x]^(n + 1), x] (* Oliver Seipel, Sep 19 2021 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) / eta(x + A)^2, n))}; /* Michael Somos, Nov 01 2008 */
  • PARI
    {a(n)=polcoeff(exp(sum(m=1,n\2+1,2*x^(2*m-1)/(1-x^(2*m-1)+x*O(x^n))/(2*m-1))),n)} /* Paul D. Hanna, Aug 06 2009 */
  • PARI
    N=66; x='x+O('x^N); gf=exp(sum(n=1,N,(sigma(2*n)-sigma(n))*x^n/n));Vec(gf) /* Joerg Arndt, Jul 30 2011 */
  • PARI
    lista(nn) = {q='q+O('q^nn); Vec(eta(q^2)/eta(q)^2)} \\ Altug Alkan, Mar 20 2018
  • SageMath
    # uses[EulerTransform from A166861]
a = BinaryRecurrenceSequence(0, 1, 1, 2)
b = EulerTransform(a)
print([b(n) for n in range(40)]) # Peter Luschny, Nov 11 2020

Formula

Euler transform of period 2 sequence [2, 1, ...]. - Michael Somos, Mar 17 2003
G.f.: Product_{m>=1} (1 + q^m)/(1 - q^m).
G.f.: 1 / (Sum_{m=-inf..inf} (-q)^(m^2)) = 1/theta_4(q).
G.f.: 1 / Product_{m>=1} (1 - q^(2*m)) * (1 - q^(2*m-1))^2.
G.f.: exp( Sum_{n>=1} 2*x^(2*n-1)/(1 - x^(2*n-1))/(2*n-1) ). - Paul D. Hanna, Aug 06 2009
G.f.: exp( Sum_{n>=1} (sigma(2*n) - sigma(n))*x^n/n ). - Joerg Arndt, Jul 30 2011
G.f.: Product_{n>=0} theta_3(q^(2^n))^(2^n). - Joerg Arndt, Aug 03 2011
A004402(n) = (-1)^n * a(n). - Michael Somos, Mar 17 2003
Expansion of eta(q^2) / eta(q)^2 in powers of q. - Michael Somos, Nov 01 2008
Expansion of 1 / phi(-q) in powers of q where phi() is a Ramanujan theta function. - Michael Somos, Nov 01 2008
Convolution inverse of A002448. - Michael Somos, Nov 01 2008
Recurrence: a(n) = 2*Sum_{m>=1} (-1)^(m+1) * a(n-m^2).
a(n) = (1/n)*Sum_{k=1..n} (sigma(2*k) - sigma(k))*a(n-k). - Vladeta Jovovic, Dec 05 2004
G.f.: Product_{i>=1} (1 + x^i)^A001511(2i) (see A000041). - Jon Perry, Jun 06 2004
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = w^4 * (u^4 + v^4) - 2 * u^2 * v^6. - Michael Somos, Nov 01 2008
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = u6^3 * (u1^2 + u3^2) - 2 * u1 * u2 * u3^3. - Michael Somos, Nov 01 2008
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = u2^3 * (u3^2 - 3 * u1^2) + 2 * u1^3 * u3 * u6. - Michael Somos, Nov 01 2008
G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 32^(-1/2) (t/i)^(-1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A106507. - Michael Somos, Nov 01 2008
a(n) = 2*A014968(n), n >= 1. - Omar E. Pol, Jan 19 2014
a(n) ~ Pi * BesselI(3/2, Pi*sqrt(n)) / (4*sqrt(2)*n^(3/4)). - Vaclav Kotesovec, Jan 11 2017
Let T(n,k) = the number of partitions of n with parts 1 through k of two kinds, T(n,0) = A000041(n), the number of partitions of n. Then a(n) = T(n,0) + T(n-1,1) + T(n-3,2) + T(n-6,3) + T(n-10,4) + T(n-15,5) + ... . Gregory L. Simay, May 29 2019
For n >= 1, a(n) = Sum_{k>=1} 2^k * A116608(n,k). - Gregory L. Simay, Jun 01 2019
Sum_{n>=1} 1/a(n) = A303662. - Amiram Eldar, Nov 15 2020
a(n) = Sum_{i=1..p(n)} 2^(d(n,i)), where d(n,i) is the number of distinct parts in the i-th partition of n. - Richard Joseph Boland, Sep 02 2021
G.f.: A(x) = exp( Sum_{n >= 1} x^n*(2 + x^n)/(n*(1 - x^(2*n))) ). - Peter Bala, Dec 23 2021
G.f. A(q) satisfies (3*A(q)/A(q^9) - 1)^3 = 9*A(q)^4/A(q^3)^4 - 1. - Paul D. Hanna, Oct 14 2024

Extensions

Minor edits by Vaclav Kotesovec, Sep 13 2014

A035359 Number of partitions-into-distinct-parts of n (A000009) is a prime.

Original entry on oeis.org

3, 4, 5, 7, 22, 70, 100, 495, 1247, 2072, 320397, 3335367, 16168775, 37472505, 52940251, 78840125, 81191852
Offset: 1

Views

Author

Olivier Gérard

Keywords

Comments

No other terms below 10^8. - Max Alekseyev, Jul 10 2015

Examples

			From _Gus Wiseman_, Jan 13 2020: (Start)
Strict partitions of a(1) = 3 through a(4) = 7:
  (3)    (4)    (5)    (7)
  (2,1)  (3,1)  (3,2)  (4,3)
                (4,1)  (5,2)
                       (6,1)
                       (4,2,1)
(End)

Links

Crossrefs

The non-strict version is A046063.
The version for powers of 2 instead of primes is A331022.
The version for factorizations instead of strict partitions is A330991.
The version for strict factorizations instead of strict partitions is A331201.
Cf. A000009, A051005, A056848, A265835.

Programs

Extensions

More terms from Eric W. Weisstein
a(12) from Max Alekseyev, Jul 04 2009
a(13)-a(14) from Giovanni Resta, Jun 05 2015, Jun 11 2015
a(15)-a(17) from Max Alekseyev, Jul 10 2015

A014968 Expansion of (1/theta_4 - 1)/2.

Original entry on oeis.org

0, 1, 2, 4, 7, 12, 20, 32, 50, 77, 116, 172, 252, 364, 520, 736, 1031, 1432, 1974, 2700, 3668, 4952, 6644, 8864, 11764, 15533, 20412, 26704, 34784, 45124, 58312, 75072, 96306, 123128, 156904, 199320, 252443, 318796, 401468, 504224, 631636, 789264, 983848, 1223532, 1518164, 1879620, 2322184, 2863040
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Let p(n) = the number of partitions of n, p(i,n) = the number of parts of the i-th partition of n, d(i,n) = the number of different parts in the i-th partition of n. Then a(n) = Sum_{i=1..p(n)} Sum_{j=1..d(i,n)} binomial(d(i,n)-1, j-1). - Thomas Wieder, May 08 2005
a(n) is the sum of the number of partitions of n-1 with two kinds of part 1 + the number of partitions of n-6 with two kinds of parts 1 through 3 + the number of partitions of n-15 with two kinds of parts 1 through 5 + ... . - Gregory L. Simay, Aug 03 2019

Examples

			G.f.: x + 2*x^2 + 4*x^3 + 7*x^4 + 12*x^5 + 20*x^6 + 32*x^7 + 50*x^8 + ...

Links

Crossrefs

Cf. A015128, A265835.
Cf. Expansion of ((Product_{n>=1} (1 - x^(k*n))/(1 - x^n)^k) - 1)/k in powers of x: this sequence (k=2), A277968 (k=3), A277974 (k=5), A160549 (k=7), A277912 (k=11).

Programs

  • Maple
    A014968 := proc(n::integer) local a,i,j,prttn,prttnlst,ZahlTeile,ZahlVerschiedenerTeile; with(combinat); a := 0; prttnlst:=partition(n); for i from 1 to nops(prttnlst) do prttn := prttnlst[i]; ZahlTeile := nops(prttn); ZahlVerschiedenerTeile:=nops(convert(prttn,multiset)); for j from 1 to ZahlVerschiedenerTeile do a := a + binomial(ZahlVerschiedenerTeile-1,j-1); od; od; print("n, a(n): ",n, a); end proc;  for n from 0 to 20 do A014968(n) end do # Thomas Wieder, May 08 2005; fixed by Vaclav Kotesovec, Dec 16 2015
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i=1, 0,
      b(n, i-1))+add(2*b(n-i*j, i-1), j=`if`(i=1, n, 1)..n/i))
    end:
a:= n-> `if`(n=0, 0, b(n$2)/2):
seq(a(n), n=0..49);  # Alois P. Heinz, Feb 10 2021
  • Mathematica
    a[ n_] := SeriesCoefficient[ (1 / EllipticTheta[ 4, 0, q] - 1) / 2, {q, 0, n}]; (* Michael Somos, Nov 03 2013 *)
(QPochhammer[x^2]/QPochhammer[x]^2-1)/2 + O[x]^40 // CoefficientList[#, x]& (* Jean-François Alcover, Nov 07 2016 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) / eta(x + A)^2 - 1 ) / 2, n))}; /* Michael Somos, Nov 03 2013 */
  • PARI
    {a(n) = if( n<0, 0, polcoeff( sum(k=1, n, x^k / (1 + x^k) * prod(j=1, k, (1 + x^j) / (1 - x^j), 1 + x * O(x^(n-k)))), n))}; /* Michael Somos, Nov 03 2013 */
  • PARI
    my(x='x+O('x^66)); concat([0],Vec(eta(x^2)/eta(x)^2-1)/2) \\ Joerg Arndt, Nov 27 2016

Formula

G.f.: Sum_{k>0} (x^k / (1 + x^k)) * Product_{j=1..k} (1 + x^j) / (1 - x^j). - Michael Somos, Nov 03 2013
2 * a(n) = A015128(n) unless n=0.
a(n) ~ exp(Pi*sqrt(n)) / (4*n) * (1 - 1/(Pi*sqrt(n))). - Vaclav Kotesovec, Nov 10 2016
G.f.: (Product_{k>=1} 1/(1-x^k))*(Sum_{k>=0} x^((2*k+1)*(k+1))/((1-x)*(1-x^2)*...*(1-x^(2*k+1)))). - Gregory L. Simay, Aug 03 2019

A285217 Indices of primes in A000712.

Original entry on oeis.org

1, 2, 70, 106, 330, 366, 370, 546, 836, 1370, 1870, 2126, 2616, 4240, 4836, 4956, 9520, 10896, 11446, 14250, 15836, 16170, 18040, 18566, 26516, 28676, 37060, 40546, 40760, 46850, 52060, 57176, 67726, 74776, 78460, 90810, 98216, 108870, 115400, 115990, 123930
Offset: 1

Views

Author

Vaclav Kotesovec, Apr 14 2017

Keywords

Examples

			70 is in the sequence because A000712(70) = 7592053897 is prime.

Links

Crossrefs

Cf. A000712, A035359, A046063, A265835.

A285219 Indices of primes in A003105.

Original entry on oeis.org

5, 6, 7, 8, 9, 11, 13, 22, 28, 29, 31, 38, 47, 53, 56, 59, 63, 64, 76, 85, 88, 91, 110, 111, 124, 135, 165, 202, 210, 214, 234, 243, 256, 262, 280, 322, 335, 346, 438, 443, 458, 463, 508, 580, 590, 696, 790, 865, 903, 951, 993, 996, 1004, 1163, 1338, 1396
Offset: 1

Views

Author

Vaclav Kotesovec, Apr 14 2017

Keywords

Examples

			28 is in the sequence because A003105(28) = 47 is prime.

Links

Crossrefs

Cf. A003105, A035359, A046063, A265835.

A295291 Indices of primes in sequence A000700.

Original entry on oeis.org

8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 20, 24, 29, 32, 35, 38, 39, 51, 56, 61, 77, 82, 88, 90, 91, 92, 107, 118, 119, 123, 139, 148, 161, 162, 166, 185, 189, 190, 194, 208, 214, 333, 346, 355, 373, 401, 402, 493, 543, 567, 578, 603, 629, 653, 665, 666, 678
Offset: 1

Views

Author

Vaclav Kotesovec, Nov 19 2017

Keywords

Examples

			51 is in the sequence because A000700(51) = 107 is prime.

Links

Crossrefs

Cf. A000700, A035359, A046063, A265835, A285217.

A299961 Numbers k such that k divides the number of overpartitions of k (A015128).

Original entry on oeis.org

1, 2, 12, 13, 22, 29, 88, 284, 370, 781, 1116, 1472, 1518, 1592, 2431, 2475, 2625, 3286, 5264, 6264, 6444, 7512, 7875, 9900, 22515, 30248, 30946, 31500, 32995, 41580, 69920, 112320, 126000, 140580, 142668, 166084, 166968, 225354, 232000, 272538, 290064, 312000
Offset: 1

Views

Author

Ilya Gutkovskiy, Feb 28 2018

Keywords

Examples

			284 is in the sequence because A015128(284) = 42480456349401075392 is divisible by 284.

Links

Crossrefs

Cf. A015128, A051177, A056848, A265835, A294086.

Extensions

More terms from Vaclav Kotesovec, Mar 02 2018
Showing 1-7 of 7 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.