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A053993 The number phi_2(n) of Frobenius partitions that allow up to 2 repetitions of an integer in a row. 7
1, 1, 3, 5, 9, 14, 24, 35, 55, 81, 120, 171, 248, 345, 486, 669, 920, 1246, 1690, 2256, 3014, 3984, 5253, 6870, 8970, 11618, 15022, 19306, 24745, 31557, 40154, 50845, 64244, 80850, 101501, 126982, 158514, 197218, 244865, 303143, 374497, 461435 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
Sum of products of multiplicities of odd parts in all partitions of n (if there are no odd parts in a partition then product of multiplicities is considered to be 1). - Vladeta Jovovic, Feb 16 2005
The sequence A077285 is the same but with multiplicities of all parts.
REFERENCES
George E. Andrews, Generalized Frobenius partitions, Memoirs of the American Mathematical Society, Number 301, May 1984.
LINKS
Brian Drake, Limits of areas under lattice paths, Discrete Math. 309 (2009), no. 12, 3936-3953. See formula (18) on page 3944.
FORMULA
Expansion of q^(1/12) * eta(q^4) * eta(q^6)^2 / (eta(q) * eta(q^2) * eta(q^3) * eta(q^12)) in powers of q. - Michael Somos, Mar 09 2011
Euler transform of period 12 sequence [ 1, 2, 2, 1, 1, 1, 1, 1, 2, 2, 1, 1, ...]. - Michael Somos, Mar 09 2011
G.f.: (Product_{k>0} (1 - x^k) * (1 - x^(12*k - 10)) * (1 - x^(12*k - 9)) * (1 - x^(12*k - 3)) * (1 - x^(12*k - 2)))^(-1). [Andrews, p. 10, equation (5.9)]
a(n) ~ exp(2*Pi*sqrt(2*n)/3) / (6*sqrt(2)*n). - Vaclav Kotesovec, Nov 28 2015
EXAMPLE
1 + x + 3*x^2 + 5*x^3 + 9*x^4 + 14*x^5 + 24*x^6 + 35*x^7 + 55*x^8 + ...
q^-1 + q^11 + 3*q^23 + 5*q^35 + 9*q^47 + 14*q^59 + 24*q^71 + 35*q^83 + ...
a(6) = 24 since the 5 partitions 6 = 5+1 = 4+2 = 3+2+1 = 2+2+2 each contribute 1, the 3 partitions 4+1+1 = 3+3 = 2+2+1+1 each contribute 2, the partition 3+1+1+1 contributes 3, the partition 2+1+1+1+1 contributes 4, and the partition 1+1+1+1+1+1 contributes 6 to give total 24 = 5*1 + 3*2 + 1*3 + 1*4 + 1*6. - Michael Somos, Mar 09 2011
MAPLE
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, b(n, i-1)
+add(b(n-i*j, i-1)*`if`(irem(i, 2)=1, j, 1), j=1..n/i)))
end:
a:= n-> b(n, n):
seq(a(n), n=0..50); # Alois P. Heinz, Jul 16 2013
MATHEMATICA
b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + Sum[b[n-i*j, i-1] * If[Mod[i, 2] == 1, j, 1], {j, 1, n/i}]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jul 15 2015, after Alois P. Heinz *)
QP = QPochhammer; s = QP[q^4] * (QP[q^6]^2 / (QP[q] * QP[q^2] * QP[q^3] * QP[q^12])) + O[q]^50; CoefficientList[s, q] (* Jean-François Alcover, Nov 09 2015, adapted from PARI *)
PROG
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^4 + A) * eta(x^6 + A)^2 / (eta(x + A) * eta(x^2 + A) * eta(x^3 + A) * eta(x^12 + A)), n))} /* Michael Somos, Mar 09 2011 */
CROSSREFS
Sequence in context: A144116 A294424 A061556 * A327562 A071155 A120695
KEYWORD
easy,nonn
AUTHOR
James A. Sellers, Apr 04 2000
STATUS
approved

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Last modified March 28 14:13 EDT 2024. Contains 371254 sequences. (Running on oeis4.)