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 1-5 of 5 results.

A298994 Expansion of Product_{n>=1} (1 + (4*x)^n)^(1/2).

Original entry on oeis.org

1, 2, 6, 52, 134, 956, 4124, 20008, 73158, 439660, 1874612, 8350808, 37583004, 169862616, 779948152, 3774085968, 15435601222, 69542934604, 329825707332, 1403190752632, 6313190864052, 29079505547912, 126937389732872, 552273916408368, 2477249228318748
Offset: 0

Views

Author

Seiichi Manyama, Jan 31 2018

Keywords

Links

Crossrefs

Cf. A271235, A298411, A298993, A303074, A370739.

Programs

  • Mathematica
    CoefficientList[Series[Sqrt[QPochhammer[-1, 4*x]/2], {x, 0, 20}], x] (* Vaclav Kotesovec, Apr 18 2018 *)

Formula

Convolution inverse of A298993.
a(n) ~ 2^(2*n - 2) * exp(Pi*sqrt(n/6)) / (3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Apr 18 2018
Sum_{k=0..n} a(k)*a(n-k) = 4^n * A000009(n). - Vaclav Kotesovec, Jun 07 2025

A271235 G.f. equals the square root of P(4*x), where P(x) is the g.f. of the partition numbers (A000041).

Original entry on oeis.org

1, 2, 14, 68, 406, 1820, 10892, 48008, 266214, 1248044, 6454116, 29642424, 156638076, 707729176, 3551518936, 16671232784, 81685862790, 375557689292, 1843995831412, 8437648295384, 40779718859796, 188104838512840, 891508943457064, 4091507664092016, 19457793452994012, 88760334081132280, 415942096027738728, 1905990594266105648, 8875964207106121784, 40416438507461834160
Offset: 0

Views

Author

Paul D. Hanna, Apr 02 2016

Keywords

Comments

More formulas and information can be derived from entry A000041.
This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = 1/2, g(n) = 4^n. - Seiichi Manyama, Apr 20 2018

Examples

			G.f.: A(x) = 1 + 2*x + 14*x^2 + 68*x^3 + 406*x^4 + 1820*x^5 + 10892*x^6 + 48008*x^7 + 266214*x^8 + 1248044*x^9 + 6454116*x^10 +...
where A(x)^2 = P(4*x).
RELATED SERIES.
P(x) = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 11*x^6 + 15*x^7 + 22*x^8 + 30*x^9 + 42*x^10 + 56*x^11 + 77*x^12 + 101*x^13 + 135*x^14 +...+ A000041(n)*x^n +...
1/A(x)^6 = 1 - 12*x + 320*x^3 - 28672*x^6 + 9437184*x^10 - 11811160064*x^15  + 57174604644352*x^21 +...+ (-1)^n*(2*n+1)*(4*x)^(n*(n+1)/2) +...

Links

Crossrefs

Cf. A298411, A298993, A298994, A370735.
Expansion of Product_{n>=1} (1 - ((b^2)*x)^n)^(-1/b): A000041 (b=1), this sequence (b=2), A271236 (b=3), A303135 (b=4), A303136 (b=5).

Programs

  • Mathematica
    nmax = 30; CoefficientList[Series[Product[1/Sqrt[1 - (4*x)^k], {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 02 2016 *)
  • PARI
    {a(n) = polcoeff( prod(k=1,n, 1/sqrt(1 - (4*x)^k +x*O(x^n))),n)}
for(n=0,30,print1(a(n),", "))
  • PARI
    {a(n) = if( n<0, 0, polcoeff( sum( k=1, sqrtint(n+1), (4*x)^(k^2) / prod(j=1, k, 1 - (4*x)^j, 1 + x*O(x^n))^2, 1)^(1/2), n))};
for(n=0,30,print1(a(n),", "))
  • PARI
    N=99; x='x+O('x^N); Vec(prod(k=1, N, 1/(1-(4*x)^k)^(1/2))) \\ Altug Alkan, Apr 20 2018

Formula

G.f.: Product_{n>=1} 1/sqrt(1 - (4*x)^n).
Sum_{k=0..n} a(k) * a(n-k) = 4^n * A000041(n), for n>=0, where A000041(n) equals the number of partitions of n.
a(n) ~ 4^(n-1) * exp(sqrt(n/3)*Pi) / (3^(3/8) * n^(7/8)). - Vaclav Kotesovec, Apr 02 2016

A303153 Expansion of Product_{n>=1} (1 - (16*x)^n)^(1/4).

Original entry on oeis.org

1, -4, -88, -992, -19360, -97152, -4296448, 4539392, -568015360, -127621120, -39357927424, 2424998313984, -38804685471744, 799759166930944, 4879962868940800, 41563181340426240, 585185165832486912, 55834295603426754560, -75535223925056208896
Offset: 0

Views

Author

Seiichi Manyama, Apr 19 2018

Keywords

Comments

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = -1/4, g(n) = 16^n.

Links

Crossrefs

Expansion of Product_{n>=1} (1 - ((b^2)*x)^n)^(1/b): A010815 (b=1), A298411 (b=2), A303152 (b=3), this sequence (b=4), A303154 (b=5).
Cf. A303124, A303135.

Programs

  • PARI
    N=66; x='x+O('x^N); Vec(prod(k=1, N, (1-(16*x)^k)^(1/4)))

A303154 Expansion of Product_{n>=1} (1 - (25*x)^n)^(1/5).

Original entry on oeis.org

1, -5, -175, -3250, -100625, -1015000, -58034375, -154171875, -22257500000, -154144921875, -6824828906250, 175448177734375, -8774446542968750, 164769756689453125, 756859169189453125, 9661555852294921875, -16148589271240234375, 81663068586871337890625
Offset: 0

Views

Author

Seiichi Manyama, Apr 19 2018

Keywords

Comments

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = -1/5, g(n) = 25^n.

Links

Crossrefs

Expansion of Product_{n>=1} (1 - ((b^2)*x)^n)^(1/b): A010815 (b=1), A298411 (b=2), A303152 (b=3), A303153 (b=4), this sequence (b=5).
Cf. A303125, A303136.

Programs

  • PARI
    N=66; x='x+O('x^N); Vec(prod(k=1, N, (1-(25*x)^k)^(1/5)))

A303152 Expansion of Product_{n>=1} (1 - (9*x)^n)^(1/3).

Original entry on oeis.org

1, -3, -36, -207, -2214, -2754, -138591, 547722, -3730293, 30138075, 133709535, 7735237479, -35284817430, 702841889322, 3056530613769, 9493893988155, 112554319443867, 3822223052352735, -3940051663965051, 250298859930263181, -551418001934739786, 1061747224529191191
Offset: 0

Views

Author

Seiichi Manyama, Apr 19 2018

Keywords

Comments

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = -1/3, g(n) = 9^n.

Links

Crossrefs

Expansion of Product_{n>=1} (1 - ((b^2)*x)^n)^(1/b): A010815 (b=1), A298411 (b=2), this sequence (b=3), A303153 (b=4), A303154 (b=5).
Cf. A271236, A303074.

Programs

  • PARI
    N=66; x='x+O('x^N); Vec(prod(k=1, N, (1-(9*x)^k)^(1/3)))
Showing 1-5 of 5 results.

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