N. Sato - cp's OEIS frontend

A145874 Number of permutations of the numbers 1, 2, ..., n such that for all 1 <= k <= n the average of the first k numbers is at least the average of all n numbers.

1, 1, 3, 7, 35, 139, 1001, 5701, 53109, 402985, 4605271

Offset: 1

N. Sato, Oct 22 2008

			For n = 3, the 3 permutations are (2,3,1), (3,1,2) and (3,2,1).

a(n) = {nbok = 0; avg = (n+1)/2; for (j = 1, n!, perm = numtoperm(n, j); ok = 1; for (k = 1, n, if (sum(j=1, k, perm[j])/k < avg, ok = 0; break;);); if (ok, nbok++);); nbok;} \\ Michel Marcus, Aug 12 2013

a(7)-a(11) from Michel Marcus, Aug 12 2013

A108932 Number of partitions of n into parts that are congruent to 1, 5 or 6 mod 8.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 3, 3, 3, 4, 5, 6, 7, 8, 10, 12, 13, 15, 18, 21, 24, 27, 31, 36, 41, 46, 52, 60, 68, 76, 86, 97, 109, 122, 136, 153, 172, 191, 212, 237, 264, 293, 325, 360, 400, 443, 488, 539, 596, 657, 723, 796, 876, 963, 1057, 1159, 1272, 1395, 1526, 1669, 1827

Offset: 0

N. Sato, Jul 20 2005

nonn

Number of partitions of n into distinct parts that are not congruent to 3 mod 4 and the number of partitions of n into odd parts such that each part which is congruent to 3 mod 4 occurs an even number of times.

{a(n)=if(n< 0, 0, polcoeff( 1/prod(k=1,n, 1-[0,1,0,0,0,1,1,0][k%8+1]*x^k, 1+x*O(x^n)), n))} /* Michael Somos, Jul 29 2005 */

G.f.: prod_{k >= 0} 1/{(1 - x^{8k + 1})(1 - x^{8k + 5})(1 - x^{8k + 6})}.

Euler transform of period 8 sequence [1, 0, 0, 0, 1, 1, 0, 0, ...]. - Michael Somos, Jul 29 2005

A093997 Number of partitions of n with an odd number of distinct Fibonacci parts.

Original entry on oeis.org

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

Offset: 0

N. Sato, May 24 2004

Alois P. Heinz, Table of n, a(n) for n = 0..10946

Cf. A000119.

F:= combinat[fibonacci]:
b:= proc(n, i, t) option remember; `if`(n=0, t, `if`(i<2, 0,
       b(n, i-1, t)+`if`(F(i)>n, 0, b(n-F(i), i-1, 1-t))))
    end:
a:= proc(n) local j; for j from ilog[(1+sqrt(5))/2](n+1)
       while F(j+1)<=n do od; b(n, j, 0)
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Jul 11 2013

Take[ CoefficientList[ Expand[ Product[1 + x^Fibonacci[k], {k, 2, 13}]/2 - Product[1 - x^Fibonacci[k], {k, 2, 13}]/2], x], 105] (* Robert G. Wilson v, May 29 2004 *)

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

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

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

Original entry on oeis.org

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

A093998 Number of partitions of n with an even number of distinct Fibonacci parts.

Original entry on oeis.org

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

Offset: 0

N. Sato, May 24 2004

Alois P. Heinz, Table of n, a(n) for n = 0..10946
F. Ardila, The coefficients of a Fibonacci power series, Fib. Quart. 42 (3) (2004), 202-204.
N. Robbins, Fibonacci partitions, Fib. Quart. 34 (4) (1996), 306-313.
J. Shallit, Robbins and Ardila meet Berstel, Arxiv preprint arXiv:2007.14930 [math.CO], 2020.

Cf. A000119, A093997.

F:= combinat[fibonacci]:
b:= proc(n, i, t) option remember; `if`(n=0, t, `if`(i<2, 0,
       b(n, i-1, t)+`if`(F(i)>n, 0, b(n-F(i), i-1, 1-t))))
    end:
a:= proc(n) local j; for j from ilog[(1+sqrt(5))/2](n+1)
       while F(j+1)<=n do od; b(n, j, 1)
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Jul 11 2013

Take[ CoefficientList[ Expand[ Product[1 + x^Fibonacci[k], {k, 2, 13}]/2 + Product[1 - x^Fibonacci[k], {k, 2, 13}]/2], x], 105] (* Robert G. Wilson v, May 29 2004 *)

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

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

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User: N. Sato

A145874 Number of permutations of the numbers 1, 2, ..., n such that for all 1 <= k <= n the average of the first k numbers is at least the average of all n numbers.

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A108932 Number of partitions of n into parts that are congruent to 1, 5 or 6 mod 8.

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A093997 Number of partitions of n with an odd number of distinct Fibonacci parts.

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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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A093998 Number of partitions of n with an even number of distinct Fibonacci parts.

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