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 1-6 of 6 results.

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

A160549 Omit first term from A160539 and divide by 7.

Original entry on oeis.org

0, 1, 5, 20, 70, 221, 646, 1772, 4614, 11490, 27537, 63808, 143514, 314279, 671872, 1405260, 2881030, 5799093, 11476452, 22357584, 42922558, 81284699, 151974124, 280739800, 512761178, 926568075, 1657448779, 2936506316, 5155349836, 8972488674, 15487146900
Offset: 0

Views

Author

N. J. A. Sloane, Nov 14 2009

Keywords

Comments

These are Watson's coefficients beta'_n on page 125.

Examples

			G.f. = x + 5*x^2 + 20*x^3 + 70*x^4 + 221*x^5 + 646*x^6 + ...

Links

Crossrefs

Cf. A160539.
Cf. Expansion of ((Product_{n>=1} (1 - x^(k*n))/(1 - x^n)^k) - 1)/k in powers of x: A014968 (k=2), A277968 (k=3), A277974 (k=5), this sequence (k=7), A277912 (k=11).

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[(Product[(1 - x^(7*j))/(1 - x^j)^7, {j, 1, nmax}] - 1)/7, {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 10 2016 *)
  • PARI
    x='x+O('x^66); concat([0],Vec(eta(x^7)/eta(x)^7-1)/7) \\ Joerg Arndt, Nov 27 2016

Formula

From Seiichi Manyama, Nov 07 2016: (Start)
a(n) = A160539(n)/7, n > 0.
G.f.: ((Product_{n>=1} (1 - x^(7*n))/(1 - x^n)^7) - 1)/7. (End)
a(n) ~ 2^(5/4) * exp(4*Pi*sqrt(2*n/7)) / (7^(13/4) * n^(9/4)). - Vaclav Kotesovec, Nov 10 2016

Extensions

Typo in definition corrected by Seiichi Manyama, Nov 07 2016

A277912 Expansion of ((Product_{n>=1} (1 - x^(11*n))/(1 - x^n)^11) - 1)/11 in powers of x.

Original entry on oeis.org

0, 1, 7, 38, 175, 714, 2653, 9139, 29563, 90650, 265401, 746142, 2023566, 5314008, 13554912, 33673525, 81654104, 193646588, 449903128, 1025532912, 2296519589, 5058078488, 10968488747, 23440057192, 49406752403, 102792264765, 211242738976, 429066735314, 861868377262, 1713014236294, 3370525567099
Offset: 0

Views

Author

Seiichi Manyama, Nov 07 2016

Keywords

Examples

			G.f. = x + 7*x^2 + 38*x^3 + 175*x^4 + 714*x^5 + 2653*x^6 + ...

Links

Crossrefs

Cf. Expansion of ((Product_{n>=1} (1 - x^(k*n))/(1 - x^n)^k) - 1)/k in powers of x: A014968 (k=2), A277968 (k=3), A277974 (k=5), A160549 (k=7), this sequence (k=11).

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[(Product[(1 - x^(11*j))/(1 - x^j)^11, {j, 1, nmax}] - 1)/11, {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 10 2016 *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ x^11] / QPochhammer[ x]^11 - 1) / 11, {x, 0, n}]; (* Michael Somos, Nov 13 2016 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^11 + A) / eta(x + A)^11 - 1) / 11, n))}; /* Michael Somos, Nov 13 2016 */
  • PARI
    x='x+O('x^66); concat([0],Vec(eta(x^11)/eta(x)^11-1)/11) \\ Joerg Arndt, Nov 27 2016

Formula

G.f.: ((Product_{n>=1} (1 - x^(11*n))/(1 - x^n)^11) - 1)/11.
a(n) ~ 5^(11/4) * exp(4*Pi*sqrt(5*n/11)) / (sqrt(2)*11^(17/4)*n^(13/4)). - Vaclav Kotesovec, Nov 10 2016

A277968 Expansion of ((Product_{n>=1} (1 - x^(3*n))/(1 - x^n)^3) - 1)/3 in powers of x.

Original entry on oeis.org

0, 1, 3, 7, 16, 33, 66, 125, 231, 412, 720, 1227, 2056, 3380, 5478, 8745, 13792, 21483, 33114, 50510, 76344, 114356, 169920, 250503, 366666, 532975, 769758, 1104847, 1576640, 2237331, 3158208, 4435502, 6199479, 8624820, 11946096, 16475880, 22630864, 30962990
Offset: 0

Views

Author

Seiichi Manyama, Nov 07 2016

Keywords

Examples

			G.f. = x + 3*x^2 + 7*x^3 + 16*x^4 + 33*x^5 + 66*x^6 + ...

Links

Crossrefs

Cf. Expansion of ((Product_{n>=1} (1 - x^(k*n))/(1 - x^n)^k) - 1)/k in powers of x: A014968 (k=2), this sequence (k=3), A277974 (k=5), A160549 (k=7), A277912 (k=11).

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[(Product[(1 - x^(3*j))/(1 - x^j)^3, {j, 1, nmax}] - 1)/3, {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 10 2016 *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ x^3] / QPochhammer[ x]^3 - 1) / 3, {x, 0, n}]; (* Michael Somos, Nov 13 2016 *)
  • PARI
    first(n)=my(x='x); concat([0], Vec((prod(k=1, n, (1-x^(3*k))/(1-x^k)^3, 1+O(x^(n+1)))-1)/3)) \\ Charles R Greathouse IV, Nov 07 2016
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^3 + A) / eta(x + A)^3 - 1) / 3, n))}; /* Michael Somos, Nov 13 2016 */

Formula

a(n) = A273845(n)/3, n > 0.
G.f.: ((Product_{n>=1} (1 - x^(3*n))/(1 - x^n)^3) - 1)/3.
a(n) ~ exp(4*Pi*sqrt(n)/3) / (27*sqrt(2)*n^(5/4)). - Vaclav Kotesovec, Nov 10 2016

A160459 Omit first term of A160458 and divide by 5.

Original entry on oeis.org

2, 13, 66, 286, 1102, 3879, 12688, 39050, 114114, 318863, 856654, 2222688, 5589916, 13668072, 32576016, 75845402, 172830788, 386088741, 846744800, 1825447086, 3872819904, 8094022001, 16679126516, 33916289400, 68106769602, 135148379654, 265177195950
Offset: 1

Views

Author

N. J. A. Sloane, Nov 13 2009

Keywords

Links

Crossrefs

Cf. A160458, A277212, A277974.

Programs

  • PARI
    x='x+O('x^66); v=Vec((eta(x^5)/eta(x)^5)^2); vector(#v-1,j,v[j+1]/5) \\ Joerg Arndt, Nov 27 2016~

Formula

a(n) = 1/5 * A160458(n) = 1/5 * Sum_{k=0..n} A277212(k)*A277212(n-k) = 2 * A277974(n) + 5 * Sum_{k=1..n-1} A277974(k)*A277974(n-k). - Seiichi Manyama, Nov 27 2016

A277992 b(n, 2) where b(n, m) is defined by expansion of ((Product_{k>=1} (1 - x^(prime(n)*k))/(1 - x^k)^prime(n)) - 1)/prime(n) in powers of x.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 10, 11, 13, 16, 17, 20, 22, 23, 25, 28, 31, 32, 35, 37, 38, 41, 43, 46, 50, 52, 53, 55, 56, 58, 65, 67, 70, 71, 76, 77, 80, 83, 85, 88, 91, 92, 97, 98, 100, 101, 107, 113, 115, 116, 118, 121, 122, 127, 130, 133, 136, 137, 140, 142, 143, 148, 155, 157
Offset: 1

Views

Author

Seiichi Manyama, Nov 07 2016

Keywords

Comments

c(n, m) is defined by expansion of (Product_{k>=1} 1/(1 - x^k)^prime(n))/prime(n) in powers of x.
b(n, 2) = c(n, 2) for n > 1.

Examples

			a(1) = b(1, 2) = A014968(2) = 2.
a(2) = b(2, 2) = A277968(2) = c(2, 2) = A000716(2)/3 = 3.
a(3) = b(3, 2) = A277974(2) = c(3, 2) = A023004(2)/5 = 4.
a(4) = b(4, 2) = A160549(2) = c(4, 2) = A023006(2)/7 = 5.
a(5) = b(5, 2) = A277912(2) = c(5, 2) = A023010(2)/11 = 7.

Crossrefs

Cf. A014968, A277968, A277974, A160549, A277912.
Expansion of Product_{k>=1} 1/(1 - x^k)^prime(n): A000712 (n=1), A000716 (n=2), A023004 (n=3), A023006 (n=4), A023010 (n=5).

Formula

a(n) = A098090(n - 1) = (prime(n) + 3)/2 for n > 1.
Showing 1-6 of 6 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.