A093998 -id:A093998 - cp's OEIS frontend

A093996 G.f.: Product_{k>=2} (1 - x^{F_k}) where F_k are the Fibonacci numbers.

1, -1, -1, 0, 1, 0, 0, 1, -1, 0, 0, 1, -1, -1, 1, 0, 0, 0, 1, -1, -1, 0, 1, 1, -1, 0, 0, 0, 0, 1, -1, -1, 0, 1, 0, 0, 1, 0, -1, -1, 1, 0, 0, 0, 0, 0, 0, 1, -1, -1, 0, 1, 0, 0, 1, -1, 0, 0, 1, -1, 0, 0, -1, 0, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, -1, 0, 1, 0, 0, 1, -1, 0, 0, 1, -1, -1, 1, 0, 0, 0, 1, -1, -1, 1, 0, 0, -1, 1, 0, 0, 1

Offset: 0

N. Sato, May 24 2004

Number of partitions of n with an even number of distinct Fibonacci parts minus the number of partitions of n with an odd number of distinct Fibonacci parts.

Every term is -1, 0 or 1.

			G.f. = 1 - x - x^2 + x^4 + x^7 - x^8 + x^11 - x^12 - x^13 + x^14 + x^18 - x^19 - x^20 + x^22 + x^23 - x^24 + x^29 - x^30 - x^31 + x^33 + x^36 - x^38 - x^39 + x^40 + x^47 - ... - _N. J. A. Sloane_, May 30 2009

T. D. Noe, Table of n, a(n) for n = 0..1000
F. Ardila, The Coefficients of a Fibonacci power series, arXiv:math/0409418 [math.CO], 2004.
Neville Robbins, Fibonacci partitions, The Fibonacci Quarterly, 34.4 (1996), pp. 306-313.
Yufei Zhao, The coefficients of a truncated Fibonacci power series, Fib. Q., 46/47 (2008/2009), 53-55.

Cf. A000045, A000119, A093997, A093998, A151661.

R:=PowerSeriesRing(Integers(), 115); Coefficients(R!( (&*[1-x^Fibonacci(j): j in [2..13]]) )); // G. C. Greubel, Dec 27 2021

Take[ CoefficientList[ Expand[ Product[1 - x^Fibonacci[k], {k, 2, 13}]], x], 105] (* Robert G. Wilson v, May 29 2004 *)
nn = 11; Take[CoefficientList[Expand[Product[1 - x^Fibonacci[n], {n, 2, nn}]], x], Fibonacci[nn+1]] (* T. D. Noe, Feb 27 2014 *)

[( product( 1-x^fibonacci(j) for j in (2..14) ) ).series(x,n+1).list()[n] for n in (0..115)] # G. C. Greubel, Dec 27 2021

Ardila gives a fast recurrence.

a(n) = A093998(n) - A093997(n).

Edited and extended by Robert G. Wilson v, May 29 2004

A339371 Number of partitions of n into an even number of Fibonacci parts (with a single type of 1).

Original entry on oeis.org

1, 0, 1, 1, 3, 2, 5, 4, 8, 7, 13, 12, 18, 18, 27, 27, 39, 38, 53, 53, 72, 73, 96, 98, 126, 128, 165, 168, 209, 216, 266, 274, 334, 345, 416, 430, 514, 533, 628, 655, 766, 797, 929, 966, 1115, 1164, 1336, 1395, 1590, 1661, 1885, 1969, 2226, 2326, 2611, 2734

Offset: 0

Ilya Gutkovskiy, Dec 02 2020

nonn

			a(7) = 4 because we have [5, 2], [3, 2, 1, 1], [2, 2, 2, 1] and [2, 1, 1, 1, 1, 1].

Index entries for sequences related to partitions

Cf. A000045, A003107, A027187, A093997, A093998, A298949, A339372.

nmax = 55; CoefficientList[Series[(1/2) (Product[1/(1 - x^Fibonacci[k]), {k, 2, 26}] + Product[1/(1 + x^Fibonacci[k]), {k, 2, 26}]), {x, 0, nmax}], x]

G.f.: (1/2) * (Product_{k>=2} 1 / (1 - x^Fibonacci(k)) + Product_{k>=2} 1 / (1 + x^Fibonacci(k))).

a(n) = (A003107(n) + A298949(n)) / 2.

A339372 Number of partitions of n into an odd number of Fibonacci parts (with a single type of 1).

Original entry on oeis.org

0, 1, 1, 2, 1, 4, 3, 6, 6, 10, 9, 15, 15, 23, 22, 32, 32, 45, 46, 62, 62, 84, 84, 110, 113, 144, 147, 185, 191, 237, 243, 299, 308, 372, 387, 462, 479, 569, 591, 695, 723, 843, 879, 1017, 1063, 1222, 1273, 1459, 1523, 1732, 1812, 2048, 2141, 2411, 2523, 2830

Offset: 0

Ilya Gutkovskiy, Dec 02 2020

nonn

			a(8) = 6 because we have [8], [5, 2, 1], [3, 3, 2], [3, 2, 1, 1, 1], [2, 2, 2, 1, 1] and [2, 1, 1, 1, 1, 1, 1].

Index entries for sequences related to partitions

Cf. A000045, A003107, A027193, A093997, A093998, A298949, A339371.

nmax = 55; CoefficientList[Series[(1/2) (Product[1/(1 - x^Fibonacci[k]), {k, 2, 26}] - Product[1/(1 + x^Fibonacci[k]), {k, 2, 26}]), {x, 0, nmax}], x]

G.f.: (1/2) * (Product_{k>=2} 1 / (1 - x^Fibonacci(k)) - Product_{k>=2} 1 / (1 + x^Fibonacci(k))).

a(n) = (A003107(n) - A298949(n)) / 2.

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A093996 G.f.: Product_{k>=2} (1 - x^{F_k}) where F_k are the Fibonacci numbers.

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A339371 Number of partitions of n into an even number of Fibonacci parts (with a single type of 1).

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A339372 Number of partitions of n into an odd number of Fibonacci parts (with a single type of 1).

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