A100818 -id:A100818 - cp's OEIS frontend

A182712 Number of 2's in the last section of the set of partitions of n.

0, 0, 1, 0, 2, 1, 4, 3, 8, 7, 15, 15, 27, 29, 48, 53, 82, 94, 137, 160, 225, 265, 362, 430, 572, 683, 892, 1066, 1370, 1640, 2078, 2487, 3117, 3725, 4624, 5519, 6791, 8092, 9885, 11752, 14263, 16922, 20416, 24167, 29007, 34254, 40921, 48213, 57345, 67409

Offset: 0

Omar E. Pol, Nov 28 2010

Essentially the same as A087787 but here a(n) is the number of 2's in the last section of n, not n-2. See also A100818.
Note that a(1)..a(11) coincide with a(2)..a(12) of A005291.
Also number of 2's in all partitions of n that do not contain 1's as a part, if n >= 1. Also 0 together with the first differences of A024786. - Omar E. Pol, Nov 13 2011
Also number of 2's in the n-th section of the set of partitions of any integer >= n. For the definition of "section" see A135010. - Omar E. Pol, Dec 01 2013

			a(6) = 4 counts the 2's in 6 = 4+2 = 2+2+2. The 2's in 6 = 3+2+1 = 2+2+1+1 = 2+1+1+1+1 do not count. - _Omar E. Pol_, Nov 13 2011
From _Omar E. Pol_, Oct 27 2012: (Start)
----------------------------------
Last section               Number
of the set of                of
partitions of 6             2's
----------------------------------
6 .......................... 0
3 + 3 ...................... 0
4 + 2 ...................... 1
2 + 2 + 2 .................. 3
.   1 ...................... 0
.       1 .................. 0
.       1 .................. 0
.           1 .............. 0
.           1 .............. 0
.               1 .......... 0
.                   1 ...... 0
---------------------------------
.   8 - 4 =                  4
.
In the last section of the set of partitions of 6 the difference between the sum of the second column and the sum of the third column is 8 - 4 = 4, the same as the number of 2's, so a(6) = 4 (see also A024786).
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Column 2 of A194812.

Cf. A005291, A087787, A100818, A135010, A138121, A182703, A182713, A182714.

Table[Count[Flatten@Cases[IntegerPartitions[n], x_ /; Last[x] != 1], 2], {n, 0, 49}] (* Robert Price, May 15 2020 *)

A182712 = lambda n: sum(list(p).count(2) for p in Partitions(n) if 1 not in p) # Omar E. Pol, Nov 13 2011

It appears that A000041(n) = a(n+1) + a(n+2), n >= 0. - Omar E. Pol, Feb 04 2012
G.f.: (x^2/(1 + x))*Product_{k>=1} 1/(1 - x^k). - Ilya Gutkovskiy, Jan 03 2017
a(n) ~ exp(Pi*sqrt(2*n/3)) / (8*sqrt(3)*n). - Vaclav Kotesovec, Jun 02 2018

A062282 Number of permutations of n elements with an even number of fixed points.

Original entry on oeis.org

1, 0, 2, 2, 16, 64, 416, 2848, 22912, 205952, 2060032, 22659328, 271913984, 3534877696, 49488295936, 742324422656, 11877190795264, 201912243453952, 3634420382302208, 69053987263479808, 1381079745270120448, 29002674650671480832, 638058842314774675456

Offset: 0

Ahmed Fares (ahmedfares(AT)my-deja.com), Jul 04 2001

nonn

Let d(n) be the number of derangements of n elements (sequence A000166) then a(n) has the recursion: a(n) = d(n) + C(n,2)*d(n-2) + C(n,4)*d(n-4) + C(n,6)*d(n-6)... = A000166(n) + A000387(n) + A000475(n) + C(n,6)*d(n-6)... The E.g.f. for a(n) is: cosh(x) * exp(-x)/(1-x) and the asymptotic expression for a(n) is: a(n) ~ n! * (1 + 1/e^2)/2 i.e., as n goes to infinity the fraction of permutations that has an even number of fixed points is about (1 + 1/e^2)/2 = 0.567667...

G. C. Greubel, Table of n, a(n) for n = 0..445

Cf. A000166, A000387, A000475.

Cf. A063083, A100818, A092295, A111752, A111753, A111723, A111724, A088336, A088506.

nn = 20; d = Exp[-x]/(1 - x); Range[0, nn]! CoefficientList[Series[Cosh[x] d, {x, 0, nn}], x] (* Geoffrey Critzer, Jan 14 2012 *)
Table[Sum[Sum[(-1)^j * n!/(j!*(2*k)!), {j, 0, n - 2*k}], {k, 0, Floor[n/2]}], {n,0,50}] (* G. C. Greubel, Aug 21 2017 *)

for(n=0,50, print1(sum(k=0,n\2, sum(j=0,n-2*k, (-1)^j*n!/(j!*(2*k)!))), ", ")) \\ G. C. Greubel, Aug 21 2017

a(n) = Sum_{k=0..[n/2]} Sum_{l=0..(n-2*k)} (-1)^l * n!/((2*k)! * l!).
More generally, e.g.f. for number of degree-n permutations with an even number of k-cycles is cosh(x^k/k)*exp(-x^k/k)/(1-x). - Vladeta Jovovic, Jan 31 2006
E.g.f.: 1/(1-x)/(x*E(0)+1), where E(k) = 1 - x^2/( x^2 + (2*k+1)*(2*k+3)/E(k+1) ); (continued fraction ). - Sergei N. Gladkovskii, Dec 29 2013
Conjecture: a(n) = Sum_{k=0..n} A008290(n, k)*A059841(k). - John Keith, Jun 30 2020

More terms from Vladeta Jovovic, Jul 05 2001

A113979 Number of compositions of n with an even number of 1's.

Original entry on oeis.org

1, 0, 2, 1, 6, 6, 20, 28, 72, 120, 272, 496, 1056, 2016, 4160, 8128, 16512, 32640, 65792, 130816, 262656, 523776, 1049600, 2096128, 4196352, 8386560, 16781312, 33550336, 67117056, 134209536, 268451840, 536854528, 1073774592, 2147450880

Offset: 0

Vladeta Jovovic, Jan 31 2006

More generally, the g.f. for the number of compositions such that part m occurs with even multiplicity is (1-x)/(1-2*x)*(1-2*x+x^m-x^(m+1))/(1-2*x+2*x^m-2*x^(m+1)). - Vladeta Jovovic, Sep 01 2007

			a(4)=6 because the compositions of 4 having an even number of 1's are 4,22,211,121,112 and 1111 (the other compositions of 4 are 31 and 13).

Index entries for linear recurrences with constant coefficients, signature (2,2,-4).

Cf. A063376, A006516, A063083, A100818, A092295, A111752, A111753, A111723, A111724, A088336, A088506.

Cf. A105422, A113980.

a:=proc(n) if n mod 2 = 0 then 2^(n-2)+2^((n-2)/2) else 2^(n-2)-2^((n-3)/2) fi end: seq(a(n),n=1..38); # Emeric Deutsch, Feb 01 2006

f[n_] := If[ EvenQ[n], 2^(n - 2) + 2^((n - 2)/2), 2^(n - 2) - 2^((n - 3)/2)]; Array[f, 34] (* Robert G. Wilson v, Feb 01 2006 *)

a(n) = n-=2; (n==-2) + 1<=0, (-1)^n << (n>>1)); \\ Kevin Ryde, May 02 2023

a(0) = 1, a(n) = 2^(n-2) + 2^((n-2)/2) if n is positive and even, otherwise a(n) = 2^(n-2) - 2^((n-3)/2).
G.f.: (1-z)*(1-z-z^2)/((1-2*z)*(1-2*z^2)). - Emeric Deutsch, Feb 03 2006
E.g.f.: (1 + exp(2*x) - sqrt(2)*sinh(x*sqrt(2)) + 2*cosh(x*sqrt(2)))/4. - Sergei N. Gladkovskii, Nov 18 2011
a(k) = (1/4)*0^k + (1/4)*2^k + (1/8)*(2-sqrt(2))*(sqrt(2))^k + (1/8)*(2+sqrt(2))*(-sqrt(2))^k. - Sergei N. Gladkovskii, Nov 18 2011

More terms from Robert G. Wilson v and Emeric Deutsch, Feb 01 2006

a(0)=1 prepended and formulas corrected by Jason Yuen, Sep 09 2024

A182722 a(n) = A005291(n+1)-A182712(n).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 3, 5, 10, 14, 26, 36, 60, 83, 128, 175, 261, 351, 504, 674, 943, 1247, 1711, 2243

Offset: 0

Omar E. Pol, Nov 28 2010, Nov 29 2010

nonn

The difference between two apparently unrelated sequences which happen to have the same initial terms. - N. J. A. Sloane, Dec 01 2010

Cf. A005291, A087787, A100818, A135010, A138121, A182703, A182712.

a(n) = A005291(n+1)-A182712(n)

A113980 Number of compositions of n with an odd number of 1's.

Original entry on oeis.org

1, 0, 3, 2, 10, 12, 36, 56, 136, 240, 528, 992, 2080, 4032, 8256, 16256, 32896, 65280, 131328, 261632, 524800, 1047552, 2098176, 4192256, 8390656, 16773120, 33558528, 67100672, 134225920, 268419072, 536887296, 1073709056, 2147516416

Offset: 1

Vladeta Jovovic, Jan 31 2006

			a(4)=2 because only the compositions 31 and 13 of 4 have an odd number of 1's (the other compositions are 4,22,211,121,112 and 1111).

Index entries for linear recurrences with constant coefficients, signature (2,2,-4).

Cf. A020522, A007582, A063083, A100818, A092295, A111752, A111753, A111723, A111724, A088336, A088506.

Cf. A105422.

a:=proc(n) if n mod 2 = 0 then 2^(n-2)-2^((n-2)/2) else 2^(n-2)+2^((n-3)/2) fi end: seq(a(n),n=1..38); # Emeric Deutsch, Feb 01 2006

f[n_] := If[EvenQ[n], 2^(n - 2) - 2^((n - 2)/2), 2^(n - 2) + 2^((n - 3)/2)]; Array[f, 34] (* Robert G. Wilson v, Feb 01 2006 *)

a(n) = 2^(n-2)-2^((n-2)/2) if n is even, else a(n) = 2^(n-2)+2^((n-3)/2).
G.f.: z(1-z)^2/[(1-2z)(1-2z^2)]. - Emeric Deutsch, Feb 03 2006
G.f.: 1 + x + Q(0), where Q(k)= 1 - 1/(2^k - 2*x*2^(2*k)/(2*x*2^k - 1/(1 + 1/(2*2^k - 8*x*2^(2*k)/(4*x*2^k + 1/Q(k+1)))))); (continued fraction). - Sergei N. Gladkovskii, May 22 2013

More terms from Robert G. Wilson v and Emeric Deutsch, Feb 01 2006

A195308 a(n) = A005291(n) + A005291(n+1).

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 43, 58, 81, 109, 150, 200, 271, 359, 481, 633, 838, 1095, 1438, 1867, 2430, 3136, 4053, 5200, 6676, 8519, 10871, 13802, 17514, 22129, 27940, 35141, 44155, 55299, 69179, 86286, 107495, 133562, 165744, 205188, 253691, 312975, 385619

Offset: 1

Omar E. Pol, Feb 03 2012

nonn

This sequence arises from A005291 in the same way as A000041 arises from A182712.

Observation: a(3)..a(13) coincide with a sequence related to Stirling's numbers from the Jordan's book.

Charles Jordan, Calculus of finite differences, Chelsea Publishing Co., 1965, chapter IV, pp. 153-155.

Cf. A000041, A005291, A087787, A100818, A182712, A194439, A195017.

a(n) = A000041(n-2), 2 <= n <= 11. - Omar E. Pol, Feb 24 2013

More terms from Amiram Eldar, May 17 2025

cp's OEIS Frontend

A182712 Number of 2's in the last section of the set of partitions of n.

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A062282 Number of permutations of n elements with an even number of fixed points.

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A113979 Number of compositions of n with an even number of 1's.

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A182722 a(n) = A005291(n+1)-A182712(n).

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A113980 Number of compositions of n with an odd number of 1's.

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A195308 a(n) = A005291(n) + A005291(n+1).

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