A000085 -id:A000085 - cp's OEIS frontend

A006123 Erroneous version of A000085.

1, 1, 2, 4, 10, 26, 75, 215

Offset: 0

dead

Computation of tree based structure as detailed in Mallows Jun 1991 letter produces A000085. - Sean A. Irvine, Mar 06 2018

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

C. L. Mallows & N. J. A. Sloane, Emails, May 1991
C. L. Mallows & N. J. A. Sloane, Emails, Jun. 1991

A066223 Bisection of A000085.

Original entry on oeis.org

1, 2, 10, 76, 764, 9496, 140152, 2390480, 46206736, 997313824, 23758664096, 618884638912, 17492190577600, 532985208200576, 17411277367391104, 606917269909048576, 22481059424730751232, 881687990282453393920, 36494410645223834692096, 1589659519990672490875904

Offset: 0

N. J. A. Sloane, Dec 19 2001

Number of tableaux on 2n elements. - Roberto E. Martinez II, Jan 09 2002
a(n) = number of ways to connect 2n points labeled 1,2,...,2n in a line with 0 or more arcs such that at most one arc leaves each point. For example, with arcs separated by dashes, a(2)=10 counts {} (no arcs), 12, 13, 14, 23, 24, 34, 12-34, 13-24, 14-23. - David Callan, Sep 18 2007
a(n) = A229223(2n,2) = A229243(2,n). - Alois P. Heinz, Sep 17 2013

S. Chowla, The asymptotic behavior of solutions of difference equations, in Proceedings of the International Congress of Mathematicians (Cambridge, MA, 1950), Vol. I, 377, Amer. Math. Soc., Providence, RI, 1952.

Alois P. Heinz, Table of n, a(n) for n = 0..200
T. Copeland, Infinitesimal Generators, the Pascal Pyramid, and the Witt and Virasoro Algebras
I. Dolinka, J. East and R. D. Gray, Motzkin monoids and partial Brauer monoids, arXiv preprint arXiv:1512.02279 [math.GR], 2015.
Michael Torpey, Semigroup congruences: computational techniques and theoretical applications, Ph.D. Thesis, University of St. Andrews (Scotland, 2019).

Cf. A066224.

a:= proc(n) option remember; `if`(n<2, n+1,
      (4*n-2)*a(n-1)-2*(n-1)*(2*n-3)*a(n-2))
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Sep 17 2013

NumberOfTableaux[2n]
a[n_] := a[n] = If[n<2, n+1, (4*n-2)*a[n-1] - 2*(n-1)*(2*n-3)*a[n-2]]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Oct 13 2014, after Alois P. Heinz *)
Table[(-2)^n HypergeometricU[-n, 1/2, -(1/2)], {n, 0, 90}] (* Emanuele Munarini, Aug 31 2017 *)

a(n)=sum(k=0,n,binomial(2*n,2*k)*prod(i=1,k,2*i-1))

a(n)=if(n<0, 0, n*=2; n!*polcoeff(exp(x+x^2/2+x*O(x^n)),n))

a(n) = sum(k=0, n, C(2n, 2*k)*(2k-1)!!). - Benoit Cloitre, May 01 2003
a(n) = n!*2^n*LaguerreL(n, -1/2, -1/2). - Vladeta Jovovic, May 10 2003
E.g.f.: cosh(x)*exp(x^2/2) (with interpolated zeros) - Paul Barry, May 26 2003
E.g.f.: exp(x/(1-2*x))/sqrt(1-2*x). - Paul Barry, Apr 12 2010
a(n) = (1/sqrt(2*pi))*Int((1+x)^(2*n)*exp(-x^2/2),x,-infinity,infinity). - Paul Barry, Apr 21 2010
Conjecture: a(n) +2*(-2*n+1)*a(n-1) +2*(n-1)*(2*n-3)*a(n-2)=0. - R. J. Mathar, Nov 24 2012
Remark: the above conjectured recurrence is true and can be obtained by the e.g.f. - Emanuele Munarini, Aug 31 2017
a(n) ~ n^n*2^(n-1/2)*exp(-n+sqrt(2*n)-1/4) * (1 + 7/(24*sqrt(2*n))). - Vaclav Kotesovec, Jun 22 2013

More terms from Roberto E. Martinez II, Jan 09 2002

A111062 Triangle T(n, k) = binomial(n, k) * A000085(n-k), 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 4, 6, 3, 1, 10, 16, 12, 4, 1, 26, 50, 40, 20, 5, 1, 76, 156, 150, 80, 30, 6, 1, 232, 532, 546, 350, 140, 42, 7, 1, 764, 1856, 2128, 1456, 700, 224, 56, 8, 1, 2620, 6876, 8352, 6384, 3276, 1260, 336, 72, 9, 1, 9496, 26200, 34380, 27840, 15960, 6552, 2100, 480, 90, 10, 1

Offset: 0

Philippe Deléham, Oct 07 2005

Triangle related to A000085.
Riordan array [exp(x(2+x)/2),x]. - Paul Barry, Nov 05 2008
Array is exp(S+S^2/2) where S is A132440 the infinitesimal generator for Pascal's triangle. T(n,k) gives the number of ways to choose a subset of {1,2,...,n} of size k and then partitioning the remaining n-k elements into sets each of size 1 or 2. Cf. A122832. - Peter Bala, May 14 2012
T(n,k) is equal to the number of R-classes (equivalently, L-classes) in the D-class consisting of all rank k elements of the partial Brauer monoid of degree n. - James East, Aug 17 2015

			Rows begin:
     1;
     1,    1;
     2,    2,    1;
     4,    6,    3,    1;
    10,   16,   12,    4,    1;
    26,   50,   40,   20,    5,    1;
    76,  156,  150,   80,   30,    6,   1;
   232,  532,  546,  350,  140,   42,   7,  1;
   764, 1856, 2128, 1456,  700,  224,  56,  8, 1;
  2620, 6876, 8352, 6384, 3276, 1260, 336, 72, 9, 1;
From _Paul Barry_, Apr 23 2009: (Start)
Production matrix is:
  1, 1,
  1, 1, 1,
  0, 2, 1, 1,
  0, 0, 3, 1, 1,
  0, 0, 0, 4, 1, 1,
  0, 0, 0, 0, 5, 1, 1,
  0, 0, 0, 0, 0, 6, 1, 1,
  0, 0, 0, 0, 0, 0, 7, 1, 1,
  0, 0, 0, 0, 0, 0, 0, 8, 1, 1 (End)
From _Peter Bala_, Feb 12 2017: (Start)
The infinitesimal generator has integer entries and begins
  0
  1  0
  1  2  0
  0  3  3  0
  0  0  6  4  0
  0  0  0 10  5  0
  0  0  0  0 15  6  0
  ...
and is the generalized exponential Riordan array [x + x^2/2!,x].(End)

Muniru A Asiru, Table of n, a(n) for n = 0..1325
Igor Dolinka, James East, Athanasios Evangelou, Des FitzGerald, Nicholas Ham, James Hyde, and Nicholas Loughlin, Enumeration of idempotents in diagram semigroups and algebras, arXiv:1408.2021 [math.GR], 2014.
Igor Dolinka, James East, Athanasios Evangelou, Des FitzGerald, Nicholas Ham, James Hyde, and Nicholas Loughlin, Enumeration of idempotents in diagram semigroups and algebras, J. Combin. Theory Ser. A 131 (2015), 119-152.
Tom Halverson and Theodore N. Jacobson, Set-partition tableaux and representations of diagram algebras, arXiv:1808.08118 [math.RT], 2018.

Cf. A000085, A005425 (row sums), A007318, A013989, A122832, A132440.

Cf. A099174, A133314, A159834 (inverse matrix).

Flat(List([0..10],n->List([0..n],k->(Factorial(n)/Factorial(k))*Sum([0..n-k],j->Binomial(j,n-k-j)/(Factorial(j)*2^(n-k-j)))))); # Muniru A Asiru, Jun 29 2018

a[n_] := Sum[(2 k - 1)!! Binomial[n, 2 k], {k, 0, n/2}]; Table[Binomial[n, k] a[n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Michael De Vlieger, Aug 20 2015, after Michael Somos at A000085 *)

def A111062_triangle(dim):
    M = matrix(ZZ,dim,dim)
    for n in (0..dim-1): M[n,n] = 1
    for n in (1..dim-1):
        for k in (0..n-1):
            M[n,k] = M[n-1,k-1]+M[n-1,k]+(k+1)*M[n-1,k+1]
    return M
A111062_triangle(9) # Peter Luschny, Sep 19 2012

Sum_{k>=0} T(m, k)*T(n, k)*k! = T(m+n, 0) = A000085(m+n).
Sum_{k=0..n} T(n, k) = A005425(n).
Apparently satisfies T(n,m) = T(n-1,m-1) + T(n-1,m) + (m+1) * T(n-1,m+1). - Franklin T. Adams-Watters, Dec 22 2005 [corrected by Werner Schulte, Feb 12 2025]
T(n,k) = (n!/k!)*Sum_{j=0..n-k} C(j,n-k-j)/(j!*2^(n-k-j)). - Paul Barry, Nov 05 2008
G.f.: 1/(1-xy-x-x^2/(1-xy-x-2x^2/(1-xy-x-3x^2/(1-xy-x-4x^2/(1-... (continued fraction). - Paul Barry, Apr 23 2009
T(n,k) = C(n,k)*Sum_{j=0..n-k} C(n-k,j)*(n-k-j-1)!! where m!!=0 if m is even. - James East, Aug 17 2015
From Tom Copeland, Jun 26 2018: (Start)
E.g.f.: exp[t*p.(x)] = exp[t + t^2/2] e^(x*t).
These polynomials (p.(x))^n = p_n(x) are an Appell sequence with the lowering and raising operators L = D and R = x + 1 + D, with D = d/dx, such that L p_n(x) = n * p_(n-1)(x) and R p_n(x) = p_(n+1)(x), so the formalism of A133314 applies here, giving recursion relations.
The transpose of the production matrix gives a matrix representation of the raising operator R.
exp(D + D^2/2) x^n= e^(D^2/2) (1+x)^n = h_n(1+x) = p_n(x) = (a. + x)^n, with (a.)^n = a_n = A000085(n) and h_n(x) the modified Hermite polynomials of A099174.
A159834 with the e.g.f. exp[-(t + t^2/2)] e^(x*t) gives the matrix inverse for this entry with the umbral inverse polynomials q_n(x), an Appell sequence with the raising operator  x - 1 - D, such that umbrally composed q_n(p.(x)) = x^n = p_n(q.(x)). (End)

Corrected by Franklin T. Adams-Watters, Dec 22 2005

10th row added by James East, Aug 17 2015

A002771 Number of terms in a skew determinant: a(n) = (A000085(n) + A081919(n))/2.

Original entry on oeis.org

1, 2, 4, 13, 41, 226, 1072, 9374, 60958, 723916, 5892536, 86402812, 837641884, 14512333928, 162925851376, 3252104882056, 41477207604872, 937014810365584, 13380460644770848, 337457467862898896, 5333575373478669136, 148532521250931168352

Offset: 1

N. J. A. Sloane

nonn

T. Muir, The expression of any bordered skew determinant as a sum of products of Pfaffians, Proc. Roy. Soc. Edinburgh, 21 (1896), 342-359.
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).

Alois P. Heinz, Table of n, a(n) for n = 1..400
T. Muir, The expression of any bordered skew determinant as a sum of products of Pfaffians, Proc. Roy. Soc. Edinburgh, 21 (1896), 342-359. [Annotated scan of pages 354-357 only]
T. Muir, The Theory of Determinants in the Historical Order of Development, 4 vols., Macmillan, NY, 1906-1923, Vol. 3, p. 282.
T. Muir, The Theory of Determinants in the Historical Order of Development, 4 vols., Macmillan, NY, 1906-1923. [Annotated scans of selected pages]

Cf. A000085, A081919, A002772.

seq(sum(binomial(n, 2*k) * doublefactorial(2*k-1) * (1+doublefactorial(2*k-1))/2, k=0..floor(n/2)), n=1..40); # Sean A. Irvine, Aug 18 2014
# second Maple program:
a:= proc(n) a(n):= `if`(n<5, [1$2, 2, 4, 13][n+1],
     ((2*n-5) *a(n-1) +(n-1)*(n^2-4*n+1) *a(n-2)
      -(2*n-5)*(n-1)*(n-2) *a(n-3))/(n-4)
      +(n-1)*(n-2)*(n-3) *(a(n-5)-a(n-4)))
    end:
seq(a(n), n=1..25);  # Alois P. Heinz, Aug 18 2014

a[n_] := Sum[Binomial[n, 2*k] * (2*k-1)!! * (1 + (2*k-1)!!) / 2, {k, 0, n/2}]; Table[a[n], {n, 1, 25}] (* Jean-François Alcover, Feb 26 2015, after Sean A. Irvine *)

def A002771(n):
    A000085 = lambda n: hypergeometric([-n/2,(1-n)/2], [], 2)
    A081919 = lambda n: hypergeometric([1/2,-n/2,(1-n)/2], [], 4)
    return ((A000085(n) + A081919(n))/2).n()
[round(A002771(n)) for n in (1..22)]  # Peter Luschny, Aug 21 2014

a(n) = Sum_{k=0..floor(n/2)} binomial(n, 2*k) * (2*k-1)!! * (1 + (2*k-1)!!) / 2. - Sean A. Irvine, Aug 18 2014
(-n+4)*a(n) +(2*n-5)*a(n-1) +(n-1)*(n^2-4*n+1)*a(n-2) -(2*n-5)*(n-1)*(n-2)*a(n-3) -(n-1)*(n-2)*(n-3)*(n-4)*a(n-4) +(n-1)*(n-2)*(n-3)*(n-4)*a(n-5)=0. - R. J. Mathar, Aug 19 2014
a(n) = (hyper2F0([-n/2,(1-n)/2],[],2)+hyper3F0([1/2,-n/2,(1-n)/2],[],4))/2. - Peter Luschny, Aug 21 2014
a(n) ~ ((-1)^n*exp(-1) + exp(1)) * n^n / (2*exp(n)). - Vaclav Kotesovec, Sep 12 2014

More terms from Sean A. Irvine, Aug 18 2014

Expanded definition from Peter Luschny, Aug 21 2014

A316330 a(n) = A000085(4*n)/2^n.

Original entry on oeis.org

1, 5, 191, 17519, 2887921, 742458253, 273315477775, 136025604432743, 87816638377854497, 71278145791452802133, 70978809757898186241439, 85023410230409691052228255, 120545000629512292505954394769, 199558149588334620585072909701981, 381323441275017330119775857868585839

Offset: 0

N. J. A. Sloane, Jul 09 2018, following a suggestion from Doron Zeilberger

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..210
S. Chowla, I. N. Herstein and W. K. Moore, On recursions connected with symmetric groups I, Canad. J. Math. 3 (1951) 328-334.
Dongsu Kim and Jang Soo Kim, A combinatorial approach to the power of 2 in the number of involutions, J. Comb. Theory Ser. A 117 (8) (2010): 1082-1094.

Cf. A000085, A316331, A316332, A316333.

A316331 a(n) = A000085(4*n+1)/2^n.

Original entry on oeis.org

1, 13, 655, 71063, 13237457, 3748521653, 1495006933759, 796798642614895, 546144645571635169, 467512355698028529821, 488384275088035513080239, 611057820865315450415912327, 901643505614430586911510015025, 1548711768835068239482321088560837

Offset: 0

N. J. A. Sloane, Jul 09 2018, following a suggestion from Doron Zeilberger

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..210
S. Chowla, I. N. Herstein and W. K. Moore, On recursions connected with symmetric groups I, Canad. J. Math. 3 (1951) 328-334.
Dongsu Kim and Jang Soo Kim, A combinatorial approach to the power of 2 in the number of involutions, J. Comb. Theory Ser. A 117 (8) (2010): 1082-1094.

Cf. A000085, A316330, A316332, A316333.

A316332 a(n) = A000085(4*n+2)/2^(n+1).

Original entry on oeis.org

1, 19, 1187, 149405, 31166057, 9670072483, 4163946939067, 2370770585582221, 1722046856020416785, 1552401874990891104371, 1699257737580930574489619, 2218555640616875773883091901, 3404174268230266459851637679353, 6062646848508401565245592651382915, 12398960005973049406349011215379703723

Offset: 0

N. J. A. Sloane, Jul 09 2018, following a suggestion from Doron Zeilberger

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..210
S. Chowla, I. N. Herstein and W. K. Moore, On recursions connected with symmetric groups I, Canad. J. Math. 3 (1951) 328-334.
Dongsu Kim and Jang Soo Kim, A combinatorial approach to the power of 2 in the number of involutions, J. Comb. Theory Ser. A 117 (8) (2010): 1082-1094.

Cf. A000085, A316330, A316331, A316333.

A316333 a(n) = A000085(4*n+3)/2^(n+2).

Original entry on oeis.org

1, 29, 2231, 323423, 75151585, 25451905333, 11799518538967, 7161375112402823, 5503252915369107329, 5217568316626716585485, 5977663757214838174587319, 8136442760259565566724537711, 12972630954295515566319694027489, 23938932303527622015634131021262757

Offset: 0

N. J. A. Sloane, Jul 09 2018, following a suggestion from Doron Zeilberger

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..210
S. Chowla, I. N. Herstein and W. K. Moore, On recursions connected with symmetric groups I, Canad. J. Math. 3 (1951) 328-334.
Dongsu Kim and Jang Soo Kim, A combinatorial approach to the power of 2 in the number of involutions, J. Comb. Theory Ser. A 117 (8) (2010): 1082-1094.

Cf. A000085, A316330, A316331, A316332.

A094070 a(n) = A000085(n) * A000110(n).

Original entry on oeis.org

1, 4, 20, 150, 1352, 15428, 203464, 3162960, 55405140, 1101298600, 24222234720, 590544046744, 15715973012248, 456341011254560, 14312979247985120, 484253161428902192, 17550722413456774848, 680244627812139042016, 28053748582811428182080, 1228896901162555453603712

Offset: 0

N. J. A. Sloane, May 01 2004

nonn

Coefficients arising in combinatorial field theory.

P. Blasiak, K. A. Penson, A. I. Solomon, A. Horzela and G. E. H. Duchamp, Some useful combinatorial formulas for bosonic operators, J. Math. Phys. 46, 052110 (2005) (6 pages).
P. Blasiak, K. A. Penson, A. I. Solomon, A. Horzela and G E. H. Duchamp, Combinatorial field theories via boson normal ordering, preprint, Apr 27 2004.

P. Blasiak, K. A. Penson, A. I. Solomon, A. Horzela and G. E. H. Duchamp, Combinatorial field theories via boson normal ordering, arXiv:quant-ph/0405103, 2004-2006. The title of this paper in the arXiv was later changed to "Some useful combinatorial formulas for bosonic operators"
A. Horzela, P. Blasiak, G. E. H. Duchamp, K. A. Penson and A. I. Solomon, A product formula and combinatorial field theory

Cf. A000085, A005425, A094071, A094072, A094073, A094074.

with(combinat): with(orthopoly): seq((I/sqrt(2))^(n+1)*H(n+1,-I/sqrt(2))*bell(n+1),n=0..17); # Emeric Deutsch, Nov 22 2004

a[n_] := Sum[StirlingS1[n+1, k] 2^k BellB[k, 1/2], {k, 0, n+1}] BellB[n+1];
Table[a[n], {n, 0, 17}] (* Jean-François Alcover, Aug 07 2018 *)

a(n) = (i/sqrt(2))^(n+1)*H(n+1, -i/sqrt(2))*Bell(n+1), where i=sqrt(-1), H(n, x) are the Hermite polynomials and Bell(n) are the Bell numbers. - Emeric Deutsch, Nov 22 2004

More terms from Ralf Stephan, Oct 14 2004

A050397 Reversion of sequence of involutions (A000085).

Original entry on oeis.org

1, -2, 4, -10, 30, -104, 392, -1568, 6520, -27976, 122944, -551680, 2518912, -11684000, 54957216, -261897024, 1263216192, -6164172608, 30416619200, -151750104800, 765364073120, -3902783995520, 20123276097920

Offset: 1

Christian G. Bower, Nov 15 1999

sign

Andrew Howroyd, Table of n, a(n) for n = 1..200
N. J. A. Sloane, Transforms
Index entries for reversions of series

Cf. A000085, A050398.

# Using function CompInv from A357588.
CompInv(23, n -> simplify(hypergeom([-n/2, (1-n)/2], [], 2))); # Peter Luschny, Oct 05 2022

seq(n)=Vec(serreverse(serlaplace(-1 + exp(x+x^2/2 + O(x*x^n))))) \\ Andrew Howroyd, May 06 2023

cp's OEIS Frontend

A006123 Erroneous version of A000085.

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A066223 Bisection of A000085.

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A111062 Triangle T(n, k) = binomial(n, k) * A000085(n-k), 0 <= k <= n.

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A002771 Number of terms in a skew determinant: a(n) = (A000085(n) + A081919(n))/2.

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A316330 a(n) = A000085(4*n)/2^n.

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A316331 a(n) = A000085(4*n+1)/2^n.

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A316332 a(n) = A000085(4*n+2)/2^(n+1).

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A316333 a(n) = A000085(4*n+3)/2^(n+2).

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A094070 a(n) = A000085(n) * A000110(n).

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