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A213598 Number of partitions of n in which no parts are multiples of 49. 1
1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231, 297, 385, 490, 627, 792, 1002, 1255, 1575, 1958, 2436, 3010, 3718, 4565, 5604, 6842, 8349, 10143, 12310, 14883, 17977, 21637, 26015, 31185, 37338, 44583, 53174, 63261, 75175, 89134, 105558, 124754, 147273, 173524 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

For n<49 we have a(n)=A000041(n), for n>=49 a(n)!=A000041(n).

In Fricke page 401, he gives the expansion sigma(omega) = q^4 + q^6 + 2q^8 + 3q^10 + 5q^12 + 7q^14 + 11q^16 + 15q^18 + ... where q = exp( Pi i omega).

REFERENCES

R. Fricke, Die elliptischen Funktionen und ihre Anwendungen, Teubner, 1922, Vol. 2, see p. 401. Eq. (49)

LINKS

Table of n, a(n) for n=0..49.

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

FORMULA

Expansion of q^(-2) * eta(q^49) / eta(q) in powers of q.

Euler transform of period 49 sequence [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, ...].

Given g.f. A(x) then B(x) = x^2 * A(x) satisfies 0 = f(B(x), B(x^2),

B(x^4)) where f(u, v, w) = u * v * w * (1 - 7*v^2) - (v - w) * (u - v) * (v^2 - u*w).

G.f. is a period 1 Fourier series which satisfies f(-1 / (49 t)) = 1 / (7 f(t)) where q = exp(2 Pi i t).

G.f.: Product_{k>0} (1 - x^(49*k)) / (1 - x^k).

a(n) ~ exp(4*Pi*sqrt(2*n)/7) / (2^(1/4) * 7^(3/2) * n^(3/4)). - Vaclav Kotesovec, Oct 14 2015

EXAMPLE

G.f. = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 11*x^6 + 15*x^7 + 22*x^8 + ...

G.f. = q^2 + q^3 + 2*q^4 + 3*q^5 + 5*q^6 + 7*q^7 + 11*q^8 + 15*q^9 + 22*q^10 + ...

MATHEMATICA

a[ n_] := SeriesCoefficient[ Product[ 1 - x^k, {k, 49, n, 49}] / Product[ 1 - x^k, {k, n}], {x, 0, n}];

a[ n_] := SeriesCoefficient[ QPochhammer[ x^49] / QPochhammer[ x], {x, 0, n}]; (* Michael Somos, May 13 2014 *)

PROG

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^49 + A) / eta(x + A), n))};

CROSSREFS

Cf. A092885, A000041.

Sequence in context: A242700 A242701 A092885 * A000041 A280662 A218027

Adjacent sequences:  A213595 A213596 A213597 * A213599 A213600 A213601

KEYWORD

nonn

AUTHOR

Michael Somos, Jun 14 2012

STATUS

approved

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Last modified May 29 09:21 EDT 2017. Contains 287245 sequences.