A280454 -id:A280454 - cp's OEIS frontend

A169975 Expansion of Product_{i>=0} (1 + x^(4*i+1)).

1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 3, 2, 0, 1, 3, 3, 1, 1, 4, 4, 1, 1, 4, 5, 2, 1, 5, 7, 3, 1, 5, 8, 5, 2, 6, 10, 6, 2, 6, 12, 9, 3, 7, 14, 11, 4, 7, 16, 15, 6, 8, 19, 18, 8, 9, 21, 23, 11, 10, 24, 27, 14, 11, 27, 34, 19, 13, 30, 39, 24, 15, 33, 47, 31, 18, 37, 54, 38

Offset: 0

N. J. A. Sloane, Aug 29 2010

nonn

Number of partitions into distinct parts of the form 4*k+1.
In general, if a > 0, b > 0, GCD(a,b) = 1 and g.f. = Product_{k>=0} (1 + x^(a*k + b)), then a(n) ~ exp(Pi*sqrt(n/(3*a))) / (2^(1 + b/a) * (3*a)^(1/4) * n^(3/4)) [Meinardus, 1954]. - Vaclav Kotesovec, Aug 26 2015
Convolution of A147599 and A169975 is A000700. - Vaclav Kotesovec, Jan 18 2017

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Günter Meinardus, Über Partitionen mit Differenzenbedingungen, Mathematische Zeitschrift (1954/55), Volume: 61, page 289-302

Cf. A000700, A261612, A280454, A280456, A280457.
Cf. A015128, A080054, A261610, A261611.
Cf. A000041, A000009, A035382, A035451.
Cf. A170966-A170975.
Cf. A262928, A147599, A281243, A281244, A281245.

nmax = 200; CoefficientList[Series[Product[(1 + x^(4*k+1)), {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 26 2015 *)
nmax = 200; poly = ConstantArray[0, nmax + 1]; poly[[1]] = 1; poly[[2]] = 1; Do[If[Mod[k, 4] == 1, Do[poly[[j + 1]] += poly[[j - k + 1]], {j, nmax, k, -1}]; ], {k, 2, nmax}]; poly (* Vaclav Kotesovec, Jan 18 2017 *)

G.f.: Sum_{n>=0} (x^(2*n^2 - n) / Product_{k=1..n} (1 - x^(4*k))). - Joerg Arndt, Mar 10 2011
G.f.: G(0)/x where G(k) = 1 - 1/(1 - 1/(1 - 1/(1+(x)^(4*k+1))/G(k+1) )); (recursively defined continued fraction, see A006950). - Sergei N. Gladkovskii, Jan 28 2013
a(n) ~ exp(Pi*sqrt(n)/(2*sqrt(3))) / (2^(7/4) * 3^(1/4) * n^(3/4)) * (1 - (3*sqrt(3)/(4*Pi) + Pi/(192*sqrt(3))) / sqrt(n)). - Vaclav Kotesovec, Aug 26 2015, extended Jan 18 2017

A281243 Expansion of Product_{k>=1} (1 + x^(5*k-1)).

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 2, 1, 0, 0, 1, 2, 1, 0, 0, 1, 3, 1, 0, 0, 2, 3, 1, 0, 0, 3, 4, 1, 0, 1, 4, 4, 1, 0, 1, 5, 5, 1, 0, 2, 7, 5, 1, 0, 3, 8, 6, 1, 0, 5, 10, 6, 1, 1, 6, 12, 7, 1, 1, 9, 14, 7, 1, 2, 11, 16, 8, 1

Offset: 0

Vaclav Kotesovec, Jan 18 2017

nonn

Convolution of this sequence and A280454 is A203776.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A262928, A147599, A281244, A281245.

Cf. A261612, A169975, A280454, A280456, A280457.

nmax = 200; CoefficientList[Series[Product[(1 + x^(5*k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 200; poly = ConstantArray[0, nmax + 1]; poly[[1]] = 1; poly[[2]] = 0; Do[If[Mod[k, 5] == 4, Do[poly[[j + 1]] += poly[[j - k + 1]], {j, nmax, k, -1}]; ], {k, 2, nmax}]; poly

a(n) ~ exp(sqrt(n/15)*Pi) / (2^(9/5)*15^(1/4)*n^(3/4)) * (1 + (Pi/(240*sqrt(15)) - 3*sqrt(15)/(8*Pi)) / sqrt(n)). - Vaclav Kotesovec, Jan 18 2017, extended Jan 24 2017

G.f.: Sum_{k>=0} x^(k*(5*k + 3)/2) / Product_{j=1..k} (1 - x^(5*j)). - Ilya Gutkovskiy, Nov 24 2020

A203776 Number of partitions of n into distinct parts 5k+1 or 5k+4.

Original entry on oeis.org

1, 1, 0, 0, 1, 1, 1, 1, 0, 1, 2, 2, 1, 1, 2, 3, 3, 2, 2, 3, 5, 5, 3, 3, 5, 7, 7, 6, 5, 7, 11, 11, 8, 8, 12, 15, 15, 13, 12, 16, 22, 22, 18, 18, 24, 30, 31, 27, 26, 33, 42, 43, 37, 37, 47, 57, 58, 53, 52, 63, 78, 80, 71, 72, 88, 103, 106, 99, 98, 116, 139, 142

Offset: 0

Reinhard Zumkeller, Jan 05 2012

nonn

Convolution of A281243 and A280454. - Vaclav Kotesovec, Jan 18 2017

			a(10) = #{9+1, 6+4} = 2;
a(20) = #{19+1, 16+4, 14+6, 11+9, 9+6+4+1} = 5.
1 + x + x^4 + x^5 + x^6 + x^7 + x^9 + 2*x^10 + 2*x^11 + x^12 + x^13 + 2*x^14 + ...
q + q^61 + q^241 + q^301 + q^361 + q^421 + q^541 + 2*q^601 + 2*q^661 + q^721 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..1000 (first 251 terms from Reinhard Zumkeller)
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A047209, A003114.

a203776 = p a047209_list where
   p _      0 = 1
   p (k:ks) m = if m < k then 0 else p ks (m - k) + p ks m

a[ n_] := SeriesCoefficient[ Product[ (1 + x^(5 k - 1)) (1 + x^(5 k - 4)), {k, Ceiling[ n / 5]}], {x, 0, n}] (* Michael Somos, Mar 23 2013 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^5] QPochhammer[ -x^4, x^5], {x, 0, n}] (* Michael Somos, Mar 23 2013 *)

{a(n) = polcoeff( prod( k=1, ceil(n / 5), (1 + x^(5*k - 1)) * (1 + x^(5*k - 4)), 1 + x * O(x^n)), n)} /* Michael Somos, Mar 23 2013 */

Expansion of f( x, x^4) / f(-x^5, -x^10) in powers of x where f() is the Ramanujan two-variable theta function. - Michael Somos, Mar 23 2013
Expansion of (-x; x^5)oo (-x^4; x^5)_oo in powers of x where (x; q)_oo is the q-Pochhammer symbol. - _Michael Somos, Mar 23 2013
Euler transform of period 10 sequence [ 1, -1, 0, 1, 0, 1, 0, -1, 1, 0, ...]. - Michael Somos, Mar 23 2013
G.f.: Product_{k>0} (1 + x^(5*k - 1)) * (1 + x^(5*k - 4)). - Michael Somos, Mar 23 2013
a(n) ~ exp(sqrt(2*n/15)*Pi) / (2*30^(1/4)*n^(3/4)) * (1 + (Pi/(60*sqrt(30)) - 3*sqrt(15/2)/(8*Pi)) / sqrt(n)). - Vaclav Kotesovec, Jan 18 2017, extended Jan 24 2017

A281245 Expansion of Product_{k>=1} (1 + x^(7*k-1)).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 1, 3, 1, 0, 0, 0, 0, 2, 3, 1, 0, 0, 0, 0, 3, 4, 1, 0, 0, 0, 1, 4, 4, 1, 0, 0, 0, 1, 5, 5, 1, 0, 0, 0, 2, 7, 5, 1, 0, 0, 0

Offset: 0

Vaclav Kotesovec, Jan 18 2017

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A262928, A147599, A281243, A281244.
Cf. A261612, A169975, A280454.
Cf. A109708, A281455, A281456, A281457, A281458, A280457, A281459.

nmax = 200; CoefficientList[Series[Product[(1 + x^(7*k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 200; poly = ConstantArray[0, nmax + 1]; poly[[1]] = 1; poly[[2]] = 0; Do[If[Mod[k, 7] == 6, Do[poly[[j + 1]] += poly[[j - k + 1]], {j, nmax, k, -1}]; ], {k, 2, nmax}]; poly

a(n) ~ exp(sqrt(n/21)*Pi) / (2^(13/7)*21^(1/4)*n^(3/4)) * (1 + (13*Pi/(336*sqrt(21)) - 3*sqrt(21)/(8*Pi)) / sqrt(n)). - Vaclav Kotesovec, Jan 18 2017, extended Jan 24 2017

G.f.: Sum_{k>=0} x^(k*(7*k + 5)/2) / Product_{j=1..k} (1 - x^(7*j)). - Ilya Gutkovskiy, Nov 24 2020

A280457 Expansion of Product_{k>=0} (1 + x^(7*k+1)).

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 1, 3, 2, 0, 0, 0, 0, 1, 3, 3, 1, 0, 0, 0, 1, 4, 4, 1, 0, 0, 0, 1, 4, 5, 2, 0, 0, 0, 1, 5, 7, 3, 0, 0, 0, 1, 5, 8, 5, 1, 0, 0, 1, 6, 10, 6, 1, 0, 0, 1, 6, 12, 9, 2, 0, 0, 1, 7, 14, 11, 3, 0, 0, 1, 7, 16, 15, 5

Offset: 0

Ilya Gutkovskiy, Jan 03 2017

nonn

Number of partitions of n into distinct parts congruent to 1 mod 7.

			a(37) = 3 because we have [36, 1], [29, 8] and [22, 15].

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
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.
Index entries for related partition-counting sequences

Cf. A000700, A016993, A169975, A261612, A280454.
Cf. A262928, A147599, A281243, A281244.
Cf. A109703, A281245, A281455, A281456, A281457, A281458, A281459.

nmax = 105; CoefficientList[Series[Product[(1 + x^(7 k + 1)), {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 200; poly = ConstantArray[0, nmax + 1]; poly[[1]] = 1; poly[[2]] = 1; Do[If[Mod[k, 7] == 1, Do[poly[[j + 1]] += poly[[j - k + 1]], {j, nmax, k, -1}]; ], {k, 2, nmax}]; poly (* Vaclav Kotesovec, Jan 18 2017 *)

G.f.: Product_{k>=0} (1 + x^(7*k+1)).

a(n) ~ exp(Pi*sqrt(n)/sqrt(21))/(2*2^(1/7)*21^(1/4)*n^(3/4)) * (1 + (13*Pi/(336*sqrt(21)) - 3*sqrt(21)/(8*Pi)) / sqrt(n)). - Ilya Gutkovskiy, Jan 03 2017, extended by Vaclav Kotesovec, Jan 24 2017

A281244 Expansion of Product_{k>=1} (1 + x^(6*k-1)).

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 2, 1, 0, 0, 0, 1, 2, 1, 0, 0, 0, 1, 3, 1, 0, 0, 0, 2, 3, 1, 0, 0, 0, 3, 4, 1, 0, 0, 1, 4, 4, 1, 0, 0, 1, 5, 5, 1, 0, 0, 2, 7, 5, 1, 0, 0, 3, 8, 6, 1, 0, 0, 5, 10, 6, 1, 0, 1

Offset: 0

Vaclav Kotesovec, Jan 18 2017

nonn

Convolution of this sequence and A280456 is A098884.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Vaclav Kotesovec, Graph - minor asymptotic term, (Pi/144 - 9/(4*Pi))/sqrt(2) = -0.49100125...

Cf. A262928, A147599, A281243, A281245.

Cf. A261612, A169975, A280454, A280456, A280457.

a:= proc(n) option remember; `if`(n=0, 1, add(add(
      [0$5, 1, 0$4, -1, 1][1+irem(d, 12)]*d, d=
       numtheory[divisors](j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Jan 18 2017

nmax = 200; CoefficientList[Series[Product[(1 + x^(6*k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 200; poly = ConstantArray[0, nmax + 1]; poly[[1]] = 1; poly[[2]] = 0; Do[If[Mod[k, 6] == 5, Do[poly[[j + 1]] += poly[[j - k + 1]], {j, nmax, k, -1}]; ], {k, 2, nmax}]; poly

a(n) ~ exp(sqrt(n/2)*Pi/3) / (2^(25/12)*sqrt(3)*n^(3/4)) * (1 + (Pi/144 - 9/(4*Pi)) / sqrt(2*n)).

G.f.: Sum_{k>=0} x^(k*(3*k + 2)) / Product_{j=1..k} (1 - x^(6*j)). - Ilya Gutkovskiy, Nov 24 2020

A280456 Expansion of Product_{k>=0} (1 + x^(6*k+1)).

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 1, 2, 1, 0, 0, 0, 1, 3, 2, 0, 0, 0, 1, 3, 3, 1, 0, 0, 1, 4, 4, 1, 0, 0, 1, 4, 5, 2, 0, 0, 1, 5, 7, 3, 0, 0, 1, 5, 8, 5, 1, 0, 1, 6, 10, 6, 1, 0, 1, 6, 12, 9, 2, 0, 1, 7, 14, 11, 3, 0, 1, 7, 16, 15, 5, 0, 1, 8, 19, 18, 7, 1, 1, 8, 21, 23, 10, 1, 1, 9, 24

Offset: 0

Ilya Gutkovskiy, Jan 03 2017

nonn

Number of partitions of n into distinct parts congruent to 1 mod 6.

Convolution of A281244 and A280456 is A098884. - Vaclav Kotesovec, Jan 18 2017

			a(32) = 3 because we have [31, 1], [25, 7] and [19, 13].

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
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.
Index entries for related partition-counting sequences

Cf. A000700, A016921, A109701, A169975, A261612, A280454, A280457.

Cf. A262928, A147599, A281243, A281244, A281245.

nmax = 105; CoefficientList[Series[Product[(1 + x^(6 k + 1)), {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 200; poly = ConstantArray[0, nmax + 1]; poly[[1]] = 1; poly[[2]] = 1; Do[If[Mod[k, 6] == 1, Do[poly[[j + 1]] += poly[[j - k + 1]], {j, nmax, k, -1}]; ], {k, 2, nmax}]; poly (* Vaclav Kotesovec, Jan 18 2017 *)

G.f.: Product_{k>=0} (1 + x^(6*k+1)).

a(n) ~ exp(Pi*sqrt(n)/(3*sqrt(2)))/(2*2^(5/12)*sqrt(3)*n^(3/4)) * (1 + (Pi/(144*sqrt(2)) - 9/(4*sqrt(2)*Pi)) / sqrt(n)). - Ilya Gutkovskiy, Jan 03 2017, extended by Vaclav Kotesovec, Jan 18 2017

A284314 Expansion of Product_{k>=0} (1 - x^(5*k+1)) in powers of x.

Original entry on oeis.org

1, -1, 0, 0, 0, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, -1, 2, -1, 0, 0, -1, 2, -1, 0, 0, -1, 3, -2, 0, 0, -1, 3, -3, 1, 0, -1, 4, -4, 1, 0, -1, 4, -5, 2, 0, -1, 5, -7, 3, 0, -1, 5, -8, 5, -1, -1, 6, -10, 6, -1, -1, 6, -12, 9, -2, -1, 7, -14, 11, -3, -1, 7, -16, 15, -5

Offset: 0

Seiichi Manyama, Mar 25 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. Product_{k>=0} (1 - x^(m*k+1)): A081362 (m=2), A284312 (m=3), A284313 (m=4), this sequence (m=5).

Cf. A280454, A284097.

CoefficientList[Series[Product[1 - x^(5k + 1), {k, 0, 100}], {x, 0, 100}], x] (* Indranil Ghosh, Mar 25 2017 *)

Vec(prod(k=0, 100, 1 - x^(5*k + 1)) + O(x^101)) \\ Indranil Ghosh, Mar 25 2017

a(n) = -(1/n)*Sum_{k=1..n} A284097(k)*a(n-k), a(0) = 1.

A281271 Expansion of Product_{k>=1} (1 + x^(5*k-2)).

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 2, 0, 1, 1, 0, 2, 0, 1, 1, 0, 3, 0, 1, 2, 0, 3, 0, 1, 3, 0, 4, 1, 1, 4, 0, 4, 1, 1, 5, 0, 5, 2, 1, 7, 0, 5, 3, 1, 8, 0, 6, 5, 1, 10, 1, 6, 6, 1, 12, 1, 7, 9, 1, 14, 2, 7, 11, 1, 16, 3, 8, 15, 1

Offset: 0

Vaclav Kotesovec, Jan 18 2017

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A219607, A281243, A281272, A280454.

nmax = 100; CoefficientList[Series[Product[(1 + x^(5*k - 2)), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 100; poly = ConstantArray[0, nmax + 1]; poly[[1]] = 1; poly[[2]] = 0; Do[If[Mod[k, 5] == 3, Do[poly[[j + 1]] += poly[[j - k + 1]], {j, nmax, k, -1}]; ], {k, 2, nmax}]; poly

a(n) ~ exp(sqrt(n/15)*Pi) / (2^(8/5)*15^(1/4)*n^(3/4)) * (1 - (3*sqrt(15)/(8*Pi) + 11*Pi/(240*sqrt(15))) / sqrt(n)). - Vaclav Kotesovec, Jan 18 2017, extended Jan 24 2017

A281272 Expansion of Product_{k>=1} (1 + x^(5*k-3)).

Original entry on oeis.org

1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 2, 0, 1, 1, 0, 2, 0, 1, 1, 0, 3, 0, 2, 1, 0, 3, 0, 3, 1, 1, 4, 0, 4, 1, 1, 4, 0, 5, 1, 2, 5, 0, 7, 1, 3, 5, 0, 8, 1, 5, 6, 1, 10, 1, 6, 6, 1, 12, 1, 9, 7, 2, 14, 1, 11, 7, 3, 16, 1, 15, 8, 5, 19, 1, 18

Offset: 0

Vaclav Kotesovec, Jan 18 2017

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A219607, A281243, A281271, A280454.

nmax = 100; CoefficientList[Series[Product[(1 + x^(5*k - 3)), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 100; poly = ConstantArray[0, nmax + 1]; poly[[1]] = 1; poly[[2]] = 0; Do[If[Mod[k, 5] == 2, Do[poly[[j + 1]] += poly[[j - k + 1]], {j, nmax, k, -1}]; ], {k, 2, nmax}]; poly

a(n) ~ exp(sqrt(n/15)*Pi) / (2^(7/5)*15^(1/4)*n^(3/4)) * (1 - (3*sqrt(15)/(8*Pi) + 11*Pi/(240*sqrt(15))) / sqrt(n)). - Vaclav Kotesovec, Jan 18 2017, extended Jan 24 2017

cp's OEIS Frontend

A169975 Expansion of Product_{i>=0} (1 + x^(4*i+1)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A281243 Expansion of Product_{k>=1} (1 + x^(5*k-1)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A203776 Number of partitions of n into distinct parts 5k+1 or 5k+4.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Formula

A281245 Expansion of Product_{k>=1} (1 + x^(7*k-1)).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A280457 Expansion of Product_{k>=0} (1 + x^(7*k+1)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A281244 Expansion of Product_{k>=1} (1 + x^(6*k-1)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A280456 Expansion of Product_{k>=0} (1 + x^(6*k+1)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A284314 Expansion of Product_{k>=0} (1 - x^(5*k+1)) in powers of x.

Views

Author

Keywords