A027998 -id:A027998 - cp's OEIS frontend

A248883 Expansion of Product_{k>=1} (1+x^k)^(k^4).

1, 1, 16, 97, 457, 2297, 11113, 52049, 235334, 1039886, 4497930, 19074006, 79418883, 325184763, 1311252535, 5212704708, 20449320159, 79231806015, 303428397505, 1149325838320, 4308477305997, 15993198330782, 58815616643170, 214383601754107, 774837953045873

Offset: 0

Vaclav Kotesovec, Mar 05 2015

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..3360 (terms 0..1000 from Vaclav Kotesovec)
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 22.

Cf. A026007, A027998, A248882, A248884.

Column k=4 of A284992.

m:=50; R:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[(1+x^k)^k^4: k in [1..m]]) )); // G. C. Greubel, Oct 31 2018

b:= proc(n) option remember; add(
      (-1)^(n/d+1)*d^5, d=numtheory[divisors](n))
    end:
a:= proc(n) option remember; `if`(n=0, 1,
      add(b(k)*a(n-k), k=1..n)/n)
    end:
seq(a(n), n=0..35);  # Alois P. Heinz, Oct 16 2017

nmax=50; CoefficientList[Series[Product[(1+x^k)^(k^4),{k,1,nmax}],{x,0,nmax}],x]

x = 'x + O('x^50); Vec(prod(k=1, 50, (1 + x^k)^(k^4))) \\ Indranil Ghosh, Apr 06 2017

a(n) ~ 31^(1/12) * exp(1/5 * (31/7)^(1/6) * 6^(2/3) * Pi * n^(5/6)) / (2^(7/6) * 3^(2/3) * 7^(1/12) * n^(7/12)).
a(0) = 1, a(n) = (1/n)*Sum_{k=1..n} A284926(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 06 2017
G.f.: exp(Sum_{k>=1} (-1)^(k+1)*x^k*(1 + 11*x^k + 11*x^(2*k) + x^(3*k))/(k*(1 - x^k)^5)). - Ilya Gutkovskiy, May 30 2018

A248884 Expansion of Product_{k>=1} (1+x^k)^(k^5).

Original entry on oeis.org

1, 1, 32, 275, 1763, 12421, 85808, 561074, 3535678, 21815897, 131733641, 778099521, 4505634324, 25635135074, 143507764032, 791243636305, 4300983535471, 23070300486656, 122213931799869, 639848848696540, 3312824859756453, 16972058378914997, 86082216143323410

Offset: 0

Vaclav Kotesovec, Mar 05 2015

nonn

In general, for m > 0, if g.f. = Product_{k>=1} (1+x^k)^(k^m), then a(n) ~ 2^(zeta(-m)) * ((1-2^(-m-1)) * Gamma(m+2) * zeta(m+2))^(1/(2*m+4)) * exp((m+2)/(m+1) * ((1-2^(-m-1)) * Gamma(m+2) * zeta(m+2))^(1/(m+2)) * n^((m+1)/(m+2))) / (sqrt(2*Pi*(m+2)) * n^((m+3)/(2*m+4))).

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 22.

Cf. A026007 (m=1), A027998 (m=2), A248882 (m=3), A248883 (m=4).

Column k=5 of A284992.

m:=50; R:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[(1+x^k)^k^5: k in [1..m]]) )); // G. C. Greubel, Oct 31 2018

b:= proc(n) option remember; add(
      (-1)^(n/d+1)*d^6, d=numtheory[divisors](n))
    end:
a:= proc(n) option remember; `if`(n=0, 1,
      add(b(k)*a(n-k), k=1..n)/n)
    end:
seq(a(n), n=0..35);  # Alois P. Heinz, Oct 16 2017

nmax=50; CoefficientList[Series[Product[(1+x^k)^(k^5),{k,1,nmax}],{x,0,nmax}],x]

m=50; x='x+O('x^m); Vec(prod(k=1, m, (1+x^k)^k^5)) \\ G. C. Greubel, Oct 31 2018

a(n) ~ (5*zeta(7))^(1/14) * 3^(2/7) * exp(zeta(7)^(1/7) * 2^(-9/7) * 3^(-3/7) * 5^(1/7) * 7^(8/7) * n^(6/7)) / (2^(163/252) * 7^(3/7) * sqrt(Pi) * n^(4/7)), where zeta(7) = A013665.

A258341 Expansion of Product_{k>=1} (1+x^k)^(k*(k+1)).

Original entry on oeis.org

1, 2, 7, 24, 65, 184, 487, 1254, 3145, 7706, 18480, 43490, 100692, 229472, 515802, 1144416, 2508948, 5439642, 11671859, 24801738, 52221911, 109013538, 225718717, 463769652, 945915199, 1915895576, 3854803572, 7706786958, 15314564282, 30255672820, 59440488874

Offset: 0

Vaclav Kotesovec, May 27 2015

nonn

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

Cf. A027998, A028377, A027999, A258342, A258343, A258344, A258345, A258346, A258347.

nmax=30; CoefficientList[Series[Product[(1+x^k)^(k*(k+1)),{k,1,nmax}],{x,0,nmax}],x]

a(n) ~ 7^(1/8) / (2^(47/24) * 15^(1/8) * n^(5/8)) * exp(2025*Zeta(3)^3 / (49*Pi^8) - 135*(15/14)^(1/4) * Zeta(3)^2 / (14*Pi^5) * n^(1/4) + 3*sqrt(15/14) * Zeta(3) / Pi^2 * sqrt(n) + 2*(14/15)^(1/4)*Pi/3 * n^(3/4)), where Zeta(3) = A002117.

A258344 Expansion of Product_{k>=1} (1+x^k)^(k*(k-1)).

Original entry on oeis.org

1, 0, 2, 6, 13, 32, 69, 160, 344, 760, 1601, 3384, 7022, 14434, 29361, 59140, 118089, 233754, 459293, 895382, 1733904, 3334914, 6374654, 12111632, 22881777, 42993244, 80362496, 149464404, 276657082, 509740278, 935046158, 1707916988, 3106810873, 5629121054

Offset: 0

Vaclav Kotesovec, May 27 2015

nonn

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

Cf. A027998, A028377, A027999, A258341, A258342, A258343, A258345, A258346, A258348.

nmax=50; CoefficientList[Series[Product[(1+x^k)^(k*(k-1)),{k,1,nmax}],{x,0,nmax}],x]

a(n) ~ 7^(1/8) / (2^(43/24) * 15^(1/8) * n^(5/8)) * exp(-2025*Zeta(3)^3 / (49*Pi^8) - 135*(15/14)^(1/4) * Zeta(3)^2 / (14*Pi^5) * n^(1/4) - 3*sqrt(15/14) * Zeta(3) / Pi^2 * sqrt(n) + 2*(14/15)^(1/4)*Pi/3 * n^(3/4)), where Zeta(3) = A002117.

A261050 Expansion of Product_{k>=1} (1+x^k)^(Fibonacci(k)).

Original entry on oeis.org

1, 1, 1, 3, 5, 10, 19, 36, 67, 127, 236, 438, 811, 1496, 2750, 5046, 9224, 16827, 30630, 55623, 100803, 182342, 329205, 593326, 1067591, 1917885, 3440207, 6162004, 11021921, 19688757, 35126020, 62590629, 111398910, 198044551, 351700332, 623918086, 1105715149

Offset: 0

Vaclav Kotesovec, Aug 08 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, Asymptotics of the Euler transform of Fibonacci numbers, arXiv:1508.01796 [math.CO], Aug 07 2015

Cf. A000045, A166861, A261051, A026007, A027998, A248882, A102866, A256142, A260916.

f:= proc(n) option remember; (<<1|1>, <1|0>>^n)[1, 2] end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
       add(binomial(f(i), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..50);  # Alois P. Heinz, Aug 08 2015

nmax=40; CoefficientList[Series[Product[(1+x^k)^Fibonacci[k],{k,1,nmax}],{x,0,nmax}],x]

a(n) ~ phi^n / (2 * sqrt(Pi) * 5^(1/8) * n^(3/4)) * exp(-1/10 + 2*5^(-1/4)*sqrt(n) + s), where s = Sum_{k>=2} (-1)^(k+1) * phi^k / ((phi^(2*k) - phi^k - 1)*k) = -0.3237251774053525012502809827680337358578568068831886835557918847... and phi = A001622 = (1+sqrt(5))/2 is the golden ratio.

G.f.: exp(Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k - x^(2*k)))). - Ilya Gutkovskiy, May 29 2018

A284992 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1+x^j)^(j^k) in powers of x.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 4, 5, 2, 1, 1, 8, 13, 8, 3, 1, 1, 16, 35, 31, 16, 4, 1, 1, 32, 97, 119, 83, 28, 5, 1, 1, 64, 275, 457, 433, 201, 49, 6, 1, 1, 128, 793, 1763, 2297, 1476, 487, 83, 8, 1, 1, 256, 2315, 6841, 12421, 11113, 4962, 1141, 142, 10, 1, 1

Offset: 0

Seiichi Manyama, Apr 07 2017

			Square array begins:
  1,  1,   1,    1,     1,      1,       1,        1, ...
  1,  1,   1,    1,     1,      1,       1,        1, ...
  1,  2,   4,    8,    16,     32,      64,      128, ...
  2,  5,  13,   35,    97,    275,     793,     2315, ...
  2,  8,  31,  119,   457,   1763,    6841,    26699, ...
  3, 16,  83,  433,  2297,  12421,   68393,   382573, ...
  4, 28, 201, 1476, 11113,  85808,  678101,  5466916, ...
  5, 49, 487, 4962, 52049, 561074, 6189117, 69540142, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0-5 give A000009, A026007, A027998, A248882, A248883, A248884.
Rows (0+1),2-3 give: A000012, A000079, A007689.
Main diagonal gives A270917.
Cf. A283272, A284993.

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1, k)*binomial(i^k, j), j=0..n/i)))
    end:
A:= (n, k)-> b(n$2, k):
seq(seq(A(n, d-n), n=0..d), d=0..14);  # Alois P. Heinz, Oct 16 2017

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0,
     Sum[b[n - i*j, i - 1, k]*Binomial[i^k, j], {j, 0, n/i}]]];
A[n_, k_] := b[n, n, k];
Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 14}] // Flatten (* Jean-François Alcover, Feb 10 2021, after Alois P. Heinz *)

G.f. of column k: Product_{j>=1} (1+x^j)^(j^k).

A294102 Expansion of Product_{k>=1} (1 + x^k)^(k(3k-1)/2).

Original entry on oeis.org

1, 1, 5, 17, 44, 127, 332, 866, 2182, 5412, 13119, 31292, 73516, 170136, 388829, 877653, 1959111, 4327221, 9464856, 20511598, 44067446, 93901142, 198539477, 416696608, 868448305, 1797890682, 3698350956, 7561361750, 15369154555, 31064311255, 62449795986, 124895635385, 248538538858, 492207649241

Offset: 0

Ilya Gutkovskiy, Nov 09 2017

nonn

Weigh transform of the pentagonal numbers (A000326).

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = -n*(3*n-1)/2, g(n) = -1. - Seiichi Manyama, Nov 14 2017

Seiichi Manyama, Table of n, a(n) for n = 0..10000
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Pentagonal Number

Cf. A000326, A027998, A028377, A294836, A294837, A294838.

nmax = 33; CoefficientList[Series[Product[(1 + x^k)^(k (3 k - 1)/2), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d^2 (3 d - 1)/2, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 33}]

G.f.: Product_{k>=1} (1 + x^k)^A000326(k).
a(n) ~ exp(-225*Zeta(3)^3 / (98*Pi^8) - 9 * 5^(5/4) * Zeta(3)^2 / (4 * 7^(5/4) * Pi^5) * n^(1/4) - (3*sqrt(5/7) * Zeta(3) / (2*Pi^2)) * sqrt(n) + (2 * (7/5)^(1/4) * Pi / 3) * n^(3/4)) * 7^(1/8) / (2^(47/24) * 5^(1/8) * n^(5/8)). - Vaclav Kotesovec, Nov 10 2017
a(0) = 1 and a(n) = (1/(2*n)) * Sum_{k=1..n} b(k)*a(n-k) where b(n) = Sum_{d|n} d^2*(3*d-1)*(-1)^(1+n/d). - Seiichi Manyama, Nov 14 2017

A294837 Expansion of Product_{k>=1} (1 + x^k)^(k(5k-3)/2).

Original entry on oeis.org

1, 1, 7, 25, 73, 236, 688, 1994, 5573, 15272, 40896, 107526, 277999, 707209, 1774067, 4390665, 10734216, 25941541, 62022609, 146793160, 344129900, 799517074, 1841734224, 4208327222, 9542121050, 21477834062, 48005313446, 106579556936, 235107392079, 515441826521, 1123360284127, 2434346065621

Offset: 0

Ilya Gutkovskiy, Nov 09 2017

nonn

Weigh transform of the heptagonal numbers (A000566).

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = -n*(5*n-3)/2, g(n) = -1. - Seiichi Manyama, Nov 14 2017

Seiichi Manyama, Table of n, a(n) for n = 0..8757
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Heptagonal Number

Cf. A000566, A027998, A028377, A294102, A294836, A294838.

nmax = 31; CoefficientList[Series[Product[(1 + x^k)^(k (5 k - 3)/2), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d^2 (5 d - 3)/2, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 31}]

G.f.: Product_{k>=1} (1 + x^k)^A000566(k).
a(n) ~ 7^(1/8) * exp(2*Pi*7^(1/4) * n^(3/4) / 3^(5/4) - 9*Zeta(3) * sqrt(3*n/7) /(2*Pi^2) - 243*Zeta(3)^2 * (3*n/7)^(1/4) / (28*Pi^5) - 2187*Zeta(3)^3 / (98*Pi^8)) / (2^(15/8) * 3^(1/8) * n^(5/8)). - Vaclav Kotesovec, Nov 10 2017
a(0) = 1 and a(n) = (1/(2*n)) * Sum_{k=1..n} b(k)*a(n-k) where b(n) = Sum_{d|n} d^2*(5*d-3)*(-1)^(1+n/d). - Seiichi Manyama, Nov 14 2017

A294838 Expansion of Product_{k>=1} (1 + x^k)^(k(3k-2)).

Original entry on oeis.org

1, 1, 8, 29, 89, 301, 915, 2763, 8040, 22910, 63776, 174174, 467448, 1233836, 3209679, 8234149, 20857621, 52206847, 129227514, 316543962, 767767628, 1844925743, 4394337797, 10379319118, 24320964976, 56557678603, 130571770387, 299357973400, 681777058604, 1542840256421, 3470045577372

Offset: 0

Ilya Gutkovskiy, Nov 09 2017

nonn

Weigh transform of the octagonal numbers (A000567).

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = -n*(3*n-2), g(n) = -1. - Seiichi Manyama, Nov 14 2017

Seiichi Manyama, Table of n, a(n) for n = 0..8270
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Octagonal Number

Cf. A000567, A027998, A028377, A294102, A294836, A294837.

nmax = 30; CoefficientList[Series[Product[(1 + x^k)^(k (3 k - 2)), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d^2 (3 d - 2), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 30}]

G.f.: Product_{k>=1} (1 + x^k)^A000567(k).
a(n) ~ exp(-1800*Zeta(3)^3 / (49*Pi^8) - (9 * 2^(3/4) * 5^(5/4) * Zeta(3)^2 / (7^(5/4)*Pi^5)) * n^(1/4) - (3*sqrt(10/7) * Zeta(3) / Pi^2) * sqrt(n) + (2*(14/5)^(1/4) * Pi/3) * n^(3/4)) * 7^(1/8) / (2^(41/24) * 5^(1/8) * n^(5/8)). - Vaclav Kotesovec, Nov 10 2017
a(0) = 1 and a(n) = (1/n) * Sum_{k=1..n} b(k)*a(n-k) where b(n) = Sum_{d|n} d^2*(3*d-2)*(-1)^(1+n/d). - Seiichi Manyama, Nov 14 2017

A261053 Expansion of Product_{k>=1} (1+x^k)^(k^k).

Original entry on oeis.org

1, 1, 4, 31, 289, 3495, 51268, 891152, 17926913, 409907600, 10499834497, 297793199060, 9262502810645, 313457634240463, 11464902463397642, 450646709610954343, 18943070964019019671, 847932498252050293971, 40266255926484893366914, 2021845081107882645459639

Offset: 0

Vaclav Kotesovec, Aug 08 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..385

Cf. A023880, A261052, A026007, A027998, A248882, A102866, A256142.

m:=20; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&*[(1+x^k)^(k^k): k in [1..(m+2)]]))); // G. C. Greubel, Nov 08 2018

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(i^i, j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..25);  # Alois P. Heinz, Aug 08 2015

nmax=20; CoefficientList[Series[Product[(1+x^k)^(k^k),{k,1,nmax}],{x,0,nmax}],x]

m=20; x='x+O('x^m); Vec(prod(k=1,m, (1+x^k)^(k^k))) \\ G. C. Greubel, Nov 08 2018

a(n) ~ n^n * (1 + exp(-1)/n + (exp(-1)/2 + 4*exp(-2))/n^2).

G.f.: exp(Sum_{k>=1} ( Sum_{d|k} (-1)^(k/d+1)*d^(d+1) ) * x^k/k). - Ilya Gutkovskiy, Nov 08 2018

cp's OEIS Frontend

A248883 Expansion of Product_{k>=1} (1+x^k)^(k^4).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A248884 Expansion of Product_{k>=1} (1+x^k)^(k^5).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A258341 Expansion of Product_{k>=1} (1+x^k)^(k*(k+1)).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A258344 Expansion of Product_{k>=1} (1+x^k)^(k*(k-1)).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A261050 Expansion of Product_{k>=1} (1+x^k)^(Fibonacci(k)).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A284992 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1+x^j)^(j^k) in powers of x.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A294102 Expansion of Product_{k>=1} (1 + x^k)^(k*(3*k-1)/2).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A294837 Expansion of Product_{k>=1} (1 + x^k)^(k*(5*k-3)/2).

Views

Author

Keywords

Comments

A294102 Expansion of Product_{k>=1} (1 + x^k)^(k(3k-1)/2).

A294837 Expansion of Product_{k>=1} (1 + x^k)^(k(5k-3)/2).

A294838 Expansion of Product_{k>=1} (1 + x^k)^(k(3k-2)).