A060632 -id:A060632 - cp's OEIS frontend

A309255 a(n) = n + 1 - Sum_{k=0..n} (Stirling1(n,k) mod 2).

0, 1, 1, 2, 3, 4, 3, 4, 7, 8, 7, 8, 9, 10, 7, 8, 15, 16, 15, 16, 17, 18, 15, 16, 21, 22, 19, 20, 21, 22, 15, 16, 31, 32, 31, 32, 33, 34, 31, 32, 37, 38, 35, 36, 37, 38, 31, 32, 45, 46, 43, 44, 45, 46, 39, 40, 49, 50, 43, 44, 45, 46, 31, 32, 63, 64, 63, 64, 65, 66, 63, 64, 69, 70, 67, 68, 69

Offset: 0

Ilya Gutkovskiy, Jul 19 2019

nonn

Number of even entries in n-th row of triangle of Stirling numbers of the first kind (A048994).

Amiram Eldar, Table of n, a(n) for n = 0..10000

Cf. A000120, A048967, A048994, A060632, A309256.

Table[n + 1 - Sum[Mod[StirlingS1[n, k], 2], {k, 0, n}], {n, 0, 76}]
nmax = 76; CoefficientList[Series[1/(1 - x)^2 - (1 + x) Product[(1 + 2 x^(2^(k + 1))), {k, 0, Floor[Log[2, nmax]] + 1}], {x, 0, nmax}], x]

a(n) = n + 1 - sum(k=0, n, stirling(n, k, 1) % 2); \\ Michel Marcus, Jul 19 2019

a(n) = n + 1 - 2^hammingweight(n\2); \\ Amiram Eldar, Jul 25 2023

G.f.: 1/(1 - x)^2 - (1 + x) * Product_{k>=0} (1 + 2*x^(2^(k+1))).

a(n) = n + 1 - 2^A000120(floor(n/2)).

A190906 a(n) = gcd(n! / floor(n/2)!^2, 3^n).

Original entry on oeis.org

1, 1, 1, 3, 3, 3, 1, 1, 1, 9, 9, 9, 3, 3, 3, 9, 9, 9, 1, 1, 1, 3, 3, 3, 1, 1, 1, 27, 27, 27, 9, 9, 9, 27, 27, 27, 3, 3, 3, 9, 9, 9, 3, 3, 3, 27, 27, 27, 9, 9, 9, 27, 27, 27, 1, 1, 1, 3, 3, 3, 1, 1, 1, 9, 9, 9, 3, 3, 3, 9, 9, 9, 1, 1, 1, 3, 3, 3, 1, 1, 1

Offset: 0

Peter Luschny, Jun 30 2011

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000

Cf. A060632.

A190906 := n -> igcd(n!/iquo(n,2)!^2,3^n): seq(A190906(n),n=0..80);

sf[n_] := With[{f = Floor[n/2]}, Pochhammer[f+1, n-f]/f!]; a[n_] := GCD[sf[n], 3^n]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Jul 29 2013 *)

a(n)=gcd(n!/(n\2)!^2, 3^n) \\ Charles R Greathouse IV, Jun 30 2011

a(n)=my(s);while(n\=3,s+=n%2);3^s \\ Charles R Greathouse IV, Jun 30 2011

a(n) = gcd(A056040(n), 3^n).
a(n) <= n. - Charles R Greathouse IV, Jun 30 2011
From Johannes W. Meijer, Jun 30 2011: (Start)
a(3*n) = a(3*n+1) = a(3*n+2) = A010684(n)*a(n) for n > 1 with a(0) = a(1) = a(2) = 1.
a(9*n+3) = a(9*n+4) = a(9*n+5) = 3*a(n).
a(9*n) = a(9*n+1) = a(9*n+2) = a(9*n+6) = a(9*n+7) = a(9*n+8) = A010690(n)*a(n). (End)

A060643 Number of conjugacy classes in the symmetric group S_n that have even number of elements.

Original entry on oeis.org

0, 0, 1, 3, 5, 7, 11, 20, 28, 38, 52, 73, 97, 127, 168, 229, 295, 381, 486, 623, 788, 994, 1247, 1571, 1954, 2428, 3002, 3710, 4557, 5588, 6826, 8347, 10141, 12306, 14879, 17973, 21633, 26007, 31177, 37334, 44579, 53166, 63253, 75167, 89126, 105542, 124738

Offset: 1

Avi Peretz (njk(AT)netvision.net.il), Apr 17 2001

nonn

The total number of conjugacy classes of S_n is the partition function p(n) (sequence A000041) and the number of conjugacy classes that have odd number of elements is given in A060632 so a(n) = A000041(n) - A060632(n) for n >= 1.

Cf. A000041, A060632.

a(n) = A000041(n) - A060632(n).

More terms from Sean A. Irvine, Dec 05 2022

A160745 First differences of A160744.

Original entry on oeis.org

3, 3, 6, 6, 6, 6, 12, 12, 6, 6, 12, 12, 12, 12, 24, 24, 6, 6, 12, 12, 12, 12, 24, 24, 12, 12, 24, 24, 24, 24, 48, 48, 6, 6, 12, 12, 12, 12, 24, 24, 12, 12, 24, 24, 24, 24, 48, 48, 12, 12, 24, 24, 24, 24, 48, 48, 24, 24, 48, 48, 48, 48, 96, 96, 6, 6, 12, 12, 12, 12, 24, 24, 12, 12, 24

Offset: 1

Omar E. Pol, May 25 2009

nonn

David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Cf. A139251, A151566, A160743, A160744.

a(n)=3*A060632(n). [From R. J. Mathar, Feb 03 2010]

Terms beyond a(7) from R. J. Mathar, Feb 03 2010

A309014 a(n) = Sum_{k=0..n} (-1)^(n-k) * (Stirling2(n,k) mod 2).

Original entry on oeis.org

1, 1, 0, 1, 1, 2, 1, 1, 2, 3, 1, 2, 3, 3, 2, 1, 3, 4, 1, 3, 4, 5, 3, 2, 5, 5, 2, 3, 5, 4, 3, 1, 4, 5, 1, 4, 5, 7, 4, 3, 7, 8, 3, 5, 8, 7, 5, 2, 7, 7, 2, 5, 7, 8, 5, 3, 8, 7, 3, 4, 7, 5, 4, 1, 5, 6, 1, 5, 6, 9, 5, 4, 9, 11, 4, 7, 11, 10, 7, 3, 10, 11, 3, 8, 11, 13, 8, 5, 13, 12, 5

Offset: 0

Ilya Gutkovskiy, Jul 06 2019

nonn

Cf. A001316, A002487, A007306, A008277, A048993, A060632, A070990.

Table[Sum[(-1)^(n - k) Mod[StirlingS2[n, k], 2], {k, 0, n}], {n, 0, 90}]
nmax = 90; CoefficientList[Series[1 + x (1 + x^3) Product[(1 + x^(2^k) + x^(2^(k + 1))), {k, 1, Floor[Log[2, nmax]] + 1}], {x, 0, nmax}], x]

a(n) = sum(k=0, n, (-1)^(n-k) * (stirling(n,k,2) % 2)); \\ Michel Marcus, Jul 06 2019

G.f.: 1 + x * (1 + x^3) * Product_{k>=1} (1 + x^(2^k) + x^(2^(k+1))).

a(0) = 1; a(2*k+1) = A002487(k+1); a(2*k+2) = A002487(k).

A335110 a(n) = Sum_{k=0..n} (Stirling1(n,k) mod 2) * k.

Original entry on oeis.org

0, 1, 3, 5, 6, 8, 18, 22, 12, 14, 30, 34, 36, 40, 84, 92, 24, 26, 54, 58, 60, 64, 132, 140, 72, 76, 156, 164, 168, 176, 360, 376, 48, 50, 102, 106, 108, 112, 228, 236, 120, 124, 252, 260, 264, 272, 552, 568, 144, 148, 300, 308, 312, 320, 648, 664, 336, 344, 696, 712, 720

Offset: 0

Ilya Gutkovskiy, May 23 2020

nonn

Cf. A000120, A007494, A060632, A087748, A087755, A335063.

Table[Sum[Mod[StirlingS1[n, k], 2] k, {k, 0, n}], {n, 0, 60}]
Table[Floor[(3 n + 1)/2] 2^(DigitCount[Floor[n/2], 2, 1] - 1), {n, 0, 60}]

a(n) = sum(k=0, n, (stirling(n,k,1) % 2) * k); \\ Michel Marcus, May 23 2020

G.f.: x * g(x) + (3/4) * x * (1 + x) * g'(x), where g(x) = Product_{k>=0} (1 + 2 * x^(2^(k + 1))).

a(n) = floor((3*n + 1)/2) * 2^(A000120(floor(n/2)) - 1).

cp's OEIS Frontend

A309255 a(n) = n + 1 - Sum_{k=0..n} (Stirling1(n,k) mod 2).

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A190906 a(n) = gcd(n! / floor(n/2)!^2, 3^n).

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A060643 Number of conjugacy classes in the symmetric group S_n that have even number of elements.

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A160745 First differences of A160744.

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A309014 a(n) = Sum_{k=0..n} (-1)^(n-k) * (Stirling2(n,k) mod 2).

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A335110 a(n) = Sum_{k=0..n} (Stirling1(n,k) mod 2) * k.

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