A220481 -id:A220481 - cp's OEIS frontend

A211609 12 times the total number of smallest parts in all partitions of n, with a(0) = 0.

0, 12, 36, 60, 120, 168, 312, 420, 684, 960, 1428, 1932, 2856, 3780, 5280, 7068, 9612, 12576, 16884, 21840, 28788, 37044, 47976, 61104, 78540, 99156, 125832, 157980, 198744, 247560, 309276, 382764, 474552, 584304, 719520, 881076, 1079244, 1314636, 1601268, 1942080, 2354016, 2842116

Offset: 0

Omar E. Pol, Feb 16 2013

nonn

The product 12spt(n) appears in the formula b(n) = 12spt(n)+(24n-1)p(n) which is mentioned in several papers (see Ono's paper, see also Garvan's papers and Garvan's slides in link section). Note that b(n) is A220481(n).

Observation: first 13 terms coincide with the differences between all terms mentioned in a table of special mock Jacobi forms and the first 13 terms of A183011. For the table see Dabholkar-Murthy-Zagier paper, appendix A.1, table of Q_M (weight 2 case), M = 6, C_M = 12. See also the table in page 46. Question: do all terms coincide?

Atish Dabholkar, Sameer Murthy, Don Zagier, Quantum Black Holes, Wall Crossing, and Mock Modular Forms, arXiv:1208.4074 [hep-th], 2012-2014.
F. G. Garvan, Congruences for Andrews' spt-function modulo powers of 5, 7 and 13
F. G. Garvan, Congruences for Andrews' spt-function modulo 32760 and extension of Atkin's Hecke-type partition congruences, arXiv:1011.1957 [math.NT], 2010, see (1.5), (2.12).
F. G. Garvan, The smallest parts partition function, slides, 2012
Ken Ono, Congruences for the Andrews spt-function, PNAS January 11, 2011 108 (2) 473-476.

Cf. A000041, A092269, A183010, A183011, A211982, A220481, A220908, A220909.

a(n) = A220481(n) - A183011(n).

a(n) = 12spt(n) = 12*A092269(n) = 6(M_2(n) - N_2(n)) = 6*A211982(n) = 6*(A220909(n) - A220908(n)), n >= 1.

cp's OEIS Frontend

A211609 12 times the total number of smallest parts in all partitions of n, with a(0) = 0.

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