A002135 -id:A002135 - cp's OEIS frontend

A059422 Difference between number of even equivalence classes and odd classes of terms in a symmetric determinant of order n.

1, 1, 0, -1, -1, 3, -2, 25, -213, 1547, -13276, 129069, -1375775, 16009741, -202184274, 2753591087, -40231298023, 627731583225, -10418193719432, 183264681827863, -3406106373633009, 66695477905719251, -1372395141298236250, 29607108539572186329

Offset: 0

N. J. A. Sloane, Jan 30 2001

sign

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 260, #12, a'_n.

Cf. A002135.

CoefficientList[Series[E^(1/2*x-1/4*x^2)*(1+x)^(1/2), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Oct 07 2013 *)

E.g.f.: exp(1/2*t-1/4*t^2)*(1+t)^(1/2)

a(n) ~ (-1)^(n+1) * n^(n-1) / (sqrt(2)*exp(n+3/4)). - Vaclav Kotesovec, Oct 07 2013

A167028 Number of terms in the expansion of the determinant of a skew-symmetric matrix of order n.

Original entry on oeis.org

0, 1, 0, 6, 0, 120, 0, 5250, 0, 395010, 0, 45197460, 0, 7299452160, 0, 1580682203100, 0, 441926274289500, 0, 154940341854097800, 0, 66565404923242024800, 0, 34389901168124209507800, 0, 21034386936107260971255000, 0, 15032296693671903309613950000, 0, 12411582569784462888618434640000, 0

Offset: 1

Pietro Majer, Oct 27 2009

If n is odd a(n)=0.

Essentially a duplicate of A002370. - N. J. A. Sloane, Oct 27 2009

			Example: the determinant of a skew symmetric matrix of order n=4 is
det(A)=A(1,2)A(1,2)A(3,4)A(3,4) + 2A(1,2)A(2,3)A(1,4)A(3,4) -2A(1,2)A(2,4)A(1,3)A(3,4)+ A(1,3)A(1,3)A(2,4)A(2,4)-2A(1,3)A(2,4)A(1,4)A(2,3)+A(1,4)A(1,4)A(2,3)A(2,3).

Cf. A002370, A167028, A002135.

for n from 1 to 20 do a[n]:=n!coeftayl( (1-x^2)^(-1/4)*exp(x^2/4),x=0,n) od;

Rest[CoefficientList[Series[(1-x^2)^(-1/4)*E^(x^2/4), {x, 0, 20}], x] * Range[0, 20]!] (* Vaclav Kotesovec, Feb 15 2015 *)

Exponential generating function: (1-x^2)^(-1/4) exp(x^2/4).

Asymptotics (for even n): a(n)= (n!/Pi)exp( (-3log(n)+1+log(2))/4 ) GAMMA(3/4) (1+O(1/n)). [corrected by Vaclav Kotesovec, Feb 15 2015]. More elegant form is a(n) ~ n! * 2^(1/4) * exp(1/4) * GAMMA(3/4) / (Pi * n^(3/4)).

A254233 Number of ways to partition the multiset consisting of n copies each of 1, 2, and 3 into n sets of size 3.

Original entry on oeis.org

1, 1, 4, 10, 25, 49, 103, 184, 331, 554, 911, 1424, 2204, 3278, 4817, 6896, 9746, 13487, 18480, 24882, 33192, 43683, 56994, 73512, 94131, 119340, 150300, 187732, 233065, 287248, 352153, 428944, 519949, 626737, 752095, 897994, 1067924, 1264241, 1491155, 1751672

Offset: 0

Tatsuru Murai, Jan 27 2015

			For n = 2, the set {1,1,2,2,3,3} can be partitioned into two sets in four ways: {{112},{233}}, {{113},{223}}, {{122},{133}}, and {{123},{123}}.

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

Cf. A002135, A254243.

Column k=3 of A257462.

G.f.: (x^12-x^11+x^10+3*x^9+5*x^8+x^7+4*x^6+x^5+5*x^4+3*x^3+x^2-x+1) / ((x^2+1)*(x^2-x+1)*(x^2+x+1)^3*(x+1)^4*(x-1)^8). - Alois P. Heinz, Apr 21 2015

Fixed definition and examples by Kellen Myers, Apr 21 2015

a(14)-a(39) from Alois P. Heinz, Apr 21 2015

A059423 Number of positive terms in a symmetric determinant of order n.

Original entry on oeis.org

1, 1, 1, 2, 8, 38, 193, 1243, 8971, 77039, 711719, 7557974, 85038784, 1075040038, 14252911927, 208885729397, 3187988600057, 53129157538097, 917244939572929, 17127419083026706, 329981266766790136, 6823452191056396526, 145118585159785011041

Offset: 0

N. J. A. Sloane, Jan 30 2001

nonn

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 260, #12, p_n.

(A002135+A059422)/2. Cf. A059424.

A059424 Number of negative terms in a symmetric determinant of order n.

Original entry on oeis.org

0, 0, 1, 3, 9, 35, 195, 1218, 9184, 75492, 724995, 7428905, 86414559, 1059030297, 14455096201, 206132138310, 3228219898080, 52501425954872, 927663133292361, 16944154401198843, 333387373140423145, 6756756713150677275, 146490980301083247291

Offset: 0

N. J. A. Sloane, Jan 30 2001

nonn

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 260, #12, p_n.

(A002135-A059422)/2. Cf. A059423.

A260338 Triangle read by rows: Cayley's numbers phi(m,n) (m,n>=0). Row m contains phi(m,0), phi(m-1,1), phi(m-2,2), ..., phi(0,m).

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 5, 6, 6, 6, 17, 23, 24, 24, 24, 73, 109, 118, 120, 120, 120, 388, 618, 690, 714, 720, 720, 720, 2461, 4096, 4686, 4926, 5016, 5040, 5040, 5040, 18155, 31133, 36308, 38688, 39768, 40200, 40320, 40320, 40320

Offset: 0

N. J. A. Sloane, Jul 30 2015

			Triangle begins:
1,
1,1,
2,2,2,
5,6,6,6,
17,23,24,24,24,
73,109,118,120,120,120,
388,618,690,714,720,720,720,
...

Alois P. Heinz, Rows n = 0..140, flattened
A. Cayley, On the number of distinct terms in a symmetrical or partially symmetrical determinant, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 9, pp. 185-190. [Annotated scanned copy]

The outer diagonals are A002135, A000142.

phi:= proc(m, n) option remember;
        `if`(n+m<2, 1, `if`(n+m=2, 2, m*phi(m-1, n)-
        `if`(n=0, (m-1)*(m-2)/2*phi(m-3, 0), -n*phi(m, n-1))))
      end:
seq(seq(phi(m-k, k), k=0..m), m=0..10);  # Alois P. Heinz, Jul 30 2015

phi[m_, n_] := phi[m, n] = If[n+m < 2, 1, If[n+m == 2, 2, m*phi[m-1, n] - If[n == 0, (m-1)*(m-2)/2*phi[m-3, 0], -n*phi[m, n-1]]]]; Table[Table[phi[ m-k, k], {k, 0, m}], {m, 0, 10}] // Flatten (* Jean-François Alcover, Feb 17 2016, after Alois P. Heinz *)

phi(0,0)=1, phi(1,0)=phi(0,1)=1, phi(2,0)=phi(1,1)=phi(0,2)=2; for m>2, phi(m,0) = m*phi(m-1,0) - (m-1)*(m-2)/2*phi(m-3,0); for m>2, n>0, phi(m,n) = m*phi(m-1,n) + n*phi(m,n-1).

More terms from Alois P. Heinz, Jul 30 2015

A002136 Matrices with 2 rows.

Original entry on oeis.org

1, 2, 6, 23, 109, 618, 4096, 31133, 267219, 2557502, 27011734, 312115953, 3916844779, 53053052462, 771450742596, 11986779006647, 198204672604489, 3475110017769282, 64396888392712366, 1257612452945760503, 25815617698822423341, 555708180579477963962, 12517189538209383465496

Offset: 3

N. J. A. Sloane

nonn

a(n) is the number of ways in which a deck with n - 1 matched pairs and two singleton cards may be dealt into n hands of two cards, assuming the order of the hands and the order of the cards in each hand is irrelevant. (See Art of Problem Solving link for proof.) - Joel B. Lewis, Sep 30 2012

			For n = 3, the a(3) = 6 ways to partition the deck {1, 1, 2, 2, 3, 4} into three pairs are {11, 22, 34}, {12, 12, 34}, {13, 14, 22}, {11, 23, 24}, {12, 13, 24} and {12, 14, 23}. - _Joel B. Lewis_, Sep 30 2012

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

John Cerkan, Table of n, a(n) for n = 3..450
Art of Problem Solving, Partitioning a deck with 2 cards in n types into pairs
P. A. MacMahon, Combinations derived from m identical sets of n different letters and their connexion with general magic squares, Proc. London Math. Soc., 17 (1917), 25-41.

/* b(n) := A002135(n) */
b(n) = if(n<3, [1,1,2][n+1], n*b(n-1) - (n-1)*(n-2)*b(n-3)/2 );
c(n) = if(n<3, [1,2][n], b(n-1) + (n-1)*b(n-2) + (n-1)*(n-2)*c(n-2) );
a(n) = c(n-2);
/* Joerg Arndt, Apr 07 2013 */

a(n+1) = A002135(n) + n*A002135(n - 1) + n*(n - 1)*a(n - 1). - Joel B. Lewis, Sep 30 2012

a(n) ~ 2^(3/2) * n^(n-2) / exp(n-3/4). - Vaclav Kotesovec, Apr 27 2015

Added more terms, Joerg Arndt, Apr 07 2013

A108704 Number of partitions of 112233...nn into n pairs.

Original entry on oeis.org

1, 1, 4, 18, 126, 1110, 12120, 156660, 2341500, 39701340, 752839920, 15785181720, 362606123880, 9055825538760, 244296192460320, 7079382509799600, 219321853964413200, 7233629128601475600, 253054306933115688000, 9358989706213886138400, 364860828050107348159200

Offset: 0

Miklos Kristof, Jun 20 2005

nonn

			Partitions of 1122 into 2 pairs: 11 22, 12 12, 12 21, 21 21 = 4 partitions so a(2)=4.

Laszlo Lovasz, Combinatorial Problems and Solutions, AMS Chelsea Publishing, American Mathematical Society.

G. C. Greubel, Table of n, a(n) for n = 0..400

Cf. A002135.

a:= n-> n! *coeff(series(exp(x*x/2)/sqrt(1-2*x), x, n+1), x, n):
seq (a(n), n=0..20);

CoefficientList[Series[E^(x*x/2)/Sqrt[1-2*x], {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Sep 26 2013 *)

my(x='x+O('x^50)); Vec(serlaplace(exp(x*x/2)/sqrt(1-2*x))) \\ G. C. Greubel, May 24 2017

E.g.f.: exp(x*x/2)/sqrt(1-2*x).
a(n) ~ 2^(n+1/2)*n^n/exp(n-1/8). - Vaclav Kotesovec, Sep 26 2013
a(n) = 2^n*(n-1/2)!*2F2((1-n)/2,-n/2;1/4 -n/2,3/4 - n/2; 1/8)/sqrt(Pi). - Benedict W. J. Irwin, May 25 2016
Conjecture: a(n)-(2*n-1)*a(n-1)-(n-1)*a(n-2)+2*(n-1)*(n-2)*a(n-3)=0. - R. J. Mathar, Jun 08 2016
From Emanuele Munarini, May 25 2022: (Start)
The exponential generating series A(t) satisfies the differential equation (1-2*t)*A'(t) = (1+t-2*t^2)*A(t), which is equivalent to the conjectured recurrence.
a(n) = Sum_{k=0..n/2} binomial(n,k)*binomial(2*n-4*k,n-2*k)*(n-k)!/2^(n-k).
(End)

cp's OEIS Frontend

A059422 Difference between number of even equivalence classes and odd classes of terms in a symmetric determinant of order n.

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A167028 Number of terms in the expansion of the determinant of a skew-symmetric matrix of order n.

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A254233 Number of ways to partition the multiset consisting of n copies each of 1, 2, and 3 into n sets of size 3.

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A059423 Number of positive terms in a symmetric determinant of order n.

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A059424 Number of negative terms in a symmetric determinant of order n.

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A260338 Triangle read by rows: Cayley's numbers phi(m,n) (m,n>=0). Row m contains phi(m,0), phi(m-1,1), phi(m-2,2), ..., phi(0,m).

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A002136 Matrices with 2 rows.

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A108704 Number of partitions of 112233...nn into n pairs.

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