cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-8 of 8 results.

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

Views

Author

N. J. A. Sloane

Keywords

References

  • 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).

Links

Crossrefs

Cf. A000085, A081919, A002772.

Programs

  • Maple
    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
  • Mathematica
    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 *)
  • Sage
    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

Formula

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

Extensions

More terms from Sean A. Irvine, Aug 18 2014
Expanded definition from Peter Luschny, Aug 21 2014

A341850 Array read by antidiagonals: T(n,m) is the number of maximum matchings in the rook graph K_n X K_m.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 2, 3, 1, 1, 3, 4, 4, 3, 1, 1, 15, 16, 72, 16, 15, 1, 1, 15, 56, 132, 132, 56, 15, 1, 1, 105, 376, 7020, 2016, 7020, 376, 105, 1, 1, 105, 1912, 17280, 44928, 44928, 17280, 1912, 105, 1, 1, 945, 17984, 1920240, 1551744, 22615200, 1551744, 1920240, 17984, 945, 1
Offset: 0

Views

Author

Andrew Howroyd, Feb 21 2021

Keywords

Comments

In the case that both m and n are odd a single vertex is not covered, otherwise the maximum matchings are perfect matchings.

Examples

			Array begins:
======================================================
n\m | 0  1   2     3       4         5           6
----+-------------------------------------------------
  0 | 1  1   1     1       1         1           1 ...
  1 | 1  1   1     3       3        15          15 ...
  2 | 1  1   2     4      16        56         376 ...
  3 | 1  3   4    72     132      7020       17280 ...
  4 | 1  3  16   132    2016     44928     1551744 ...
  5 | 1 15  56  7020   44928  22615200   243319680 ...
  6 | 1 15 376 17280 1551744 243319680 61903180800 ...
  ...

Links

Crossrefs

Rows n=1..4 are A133221(n+1), A081919, A341851, A341852.
Main diagonal is A289197.
Cf. A270227 (matchings), A286070, A341847 (maximal matchings).

Formula

T(n,m) = T(m,n).

A081920 Expansion of exp(2x)/sqrt(1-x^2).

Original entry on oeis.org

1, 2, 5, 14, 49, 202, 1069, 6470, 48353, 391058, 3767029, 37936318, 445650385, 5359634906, 74198053661, 1036667808758, 16516851030721, 262805595346210, 4735033850606437, 84510767762583662, 1698609728377283441
Offset: 0

Views

Author

Paul Barry, Apr 01 2003

Keywords

Comments

Binomial transform of A081919

Links

Crossrefs

Cf. A081921.

Programs

  • Maple
    f:= gfun:-rectoproc({a(n) -2*a(n-1) -(n-1)^2*a(n-2) +2*(n-1)*(n-2)*a(n-3)=0, a(0)=1,a(1)=2,a(2)=5},a(n),remember):
map(f, [$0..30]); # Robert Israel, Feb 19 2018
  • Mathematica
    CoefficientList[Series[E^(2*x)/Sqrt[1-x^2], {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Feb 04 2014 *)

Formula

E.g.f. exp(2x)/sqrt(1-x^2).
Conjecture: a(n) -2*a(n-1) -(n-1)^2*a(n-2) +2*(n-1)*(n-2)*a(n-3)=0. - R. J. Mathar, Nov 24 2012
Conjecture confirmed using d.e. (x^2-1)*y' + (-2*x^2+x+2)*y = 0 satisfied by the E.g.f. - Robert Israel, Feb 19 2018
a(n) ~ n^n * (exp(2)+(-1)^n*exp(-2)) / exp(n). - Vaclav Kotesovec, Feb 04 2014

A270229 Number of matchings in the 2 X n rook graph P_2 X K_n.

Original entry on oeis.org

1, 2, 7, 32, 193, 1382, 11719, 112604, 1221889, 14639786, 192949639, 2760749048, 42732172993, 709490574158, 12596398359367, 237750425419508, 4757710386662401, 100516614496518866, 2236829315345704711, 52262526676903613264, 1279512810244450887361
Offset: 0

Views

Author

Andrew Howroyd, Mar 13 2016

Keywords

Comments

Sequence extended to n=0 using closed form. (binomial transform of A111883)

Links

Crossrefs

Cf. A270227, A270228, A000085, A081919 (perfect matchings).

Programs

  • Mathematica
    a[n_] := Sum[Binomial[n, k]*Abs[HermiteH[k, I/Sqrt[2]]]^2/2^k, {k, 0, n}];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Oct 01 2017 *)
CoefficientList[Series[E^((2-x)*x/(1-x)) / Sqrt[1-x^2], {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Oct 01 2017 *)

Formula

Binomial transform of A111883.
From Vaclav Kotesovec, Oct 01 2017: (Start)
a(n) = (n+1)*a(n-1) + (n-1)^2*a(n-2) - (n-2)*(n-1)^2*a(n-3) + (n-3)*(n-2)*(n-1)*a(n-4).
E.g.f.: exp((2-x)*x/(1-x)) / sqrt(1-x^2).
a(n) ~ exp(1/2 + 2*sqrt(n) - n) * n^n / 2.
(End)

A281433 Number of maximal matchings in the 2 X n rook graph.

Original entry on oeis.org

1, 1, 2, 10, 40, 296, 1576, 15352, 104000, 1276480, 10556416, 156843776, 1533722752, 26777626240, 302395339520, 6068829396736, 77740741758976, 1763457842941952, 25267740818452480, 639308368122204160, 10131932297407840256, 282891828731667890176
Offset: 0

Views

Author

Andrew Howroyd, Oct 05 2017

Keywords

Links

Crossrefs

Row n=2 of A341847.
Cf. A081919 (perfect matchings), A270229, A289198.

Programs

  • Mathematica
    a[n_] := Sum[(2*k-1)!!^2 * Binomial[n, 2*k] * (1 + 2*k*(n-2*k)), {k, 0, n/2} ]; Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Nov 01 2017, after Andrew Howroyd *)
  • PARI
    a(n) = sum(k=0, n\2, ((2*k)!/(2^k*k!))^2 * binomial(n,2*k) * (1 + 2*k*(n-2*k)));

Formula

a(n) = Sum_{k=0..n/2} (2*k-1)!!^2 * binomial(n,2*k) * (1 + 2*k*(n-2*k)).

A291482 Expansion of e.g.f. arcsin(x)*exp(x).

Original entry on oeis.org

0, 1, 2, 4, 8, 24, 80, 456, 2368, 20352, 139648, 1577984, 13327360, 185992832, 1860708096, 30882985472, 356724338688, 6860887896064, 89815091306496, 1963843714723840, 28724760194564096, 703639672161697792, 11370790299166343168, 308435832182144040960, 5456591088206554333184, 162354575283061816197120
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 24 2017

Keywords

Examples

			E.g.f.: A(x) = x/1! + 2*x^2/2! + 4*x^3/3! + 8*x^4/4! + 24*x^5/5! + ...

Crossrefs

Cf. A001818, A009545, A012316, A081919 (first differences).

Programs

  • Maple
    a:=series(arcsin(x)*exp(x),x=0,26): seq(n!*coeff(a,x,n),n=0..25); # Paolo P. Lava, Mar 27 2019
  • Mathematica
    nmax = 25; Range[0, nmax]! CoefficientList[Series[ArcSin[x] Exp[x], {x, 0, nmax}], x]
nmax = 25; Range[0, nmax]! CoefficientList[Series[Exp[x] x Sqrt[1 - x^2]/(1 + ContinuedFractionK[-2 x^2 Floor[(k + 1)/2] (2 Floor[(k + 1)/2] - 1), 2 k + 1, {k, 1, nmax}]), {x, 0, nmax}], x]
nmax = 25; Range[0, nmax]! CoefficientList[Series[Sum[(x^(2 k + 1) Pochhammer[1/2, k])/(k! + 2 k k!), {k, 0, Infinity}] Exp[x], {x, 0, nmax}], x]
Table[Sum[Binomial[n,2k+1]Binomial[2k,k] (2k)!/4^k,{k,0,(n-1)/2}],{n,0,12}] (* Emanuele Munarini, Dec 17 2017 *)
  • Maxima
    makelist(sum(binomial(n,2*k+1)*binomial(2*k,k)*(2*k)!/4^k,k,0,floor((n-1)/2)),n,0,12); /* Emanuele Munarini, Dec 17 2017 */
  • PARI
    x='x+O('x^99); concat(0, Vec(serlaplace(asin(x)*exp(x)))) \\ Altug Alkan, Dec 17 2017

Formula

E.g.f.: exp(x)*x*sqrt(1 - x^2)/(1 - 1*2*x^2/(3 - 1*2*x^2/(5 - 3*4*x^2/(7 - 3*4*x^2/(9 - ...))))), a continued fraction.
a(n) ~ (exp(2) - (-1)^n) * n^(n-1) / exp(n+1). - Vaclav Kotesovec, Aug 26 2017
From Emanuele Munarini, Dec 17 2017: (Start)
a(n) = Sum_{k=0..(n-1)/2} binomial(n,2*k+1)*binomial(2*k,k)* (2k)!/4^k.
a(n+4) - 2*a(n+3) - (n^2+4*n+3)*a(n+2) + (n+2)*(2*n+3)*a(n+1) - (n+1)*(n+2)*a(n) = 0. (End)

A081922 Expansion of exp(4x)/sqrt(1-x^2).

Original entry on oeis.org

1, 4, 17, 76, 361, 1844, 10321, 64348, 453329, 3619684, 32666161, 329434604, 3677682937, 44901581716, 595567550321, 8505627039484, 130307878338721, 2126927187154628, 36912563369550289, 677277819029706124
Offset: 0

Views

Author

Paul Barry, Apr 01 2003

Keywords

Comments

Binomial transform of A081921
Generally, if e.g.f. = exp(p*x)/sqrt(1-x^2), then a(n) ~ n^n * (exp(p)+(-1)^n*exp(-p)) / exp(n). - Vaclav Kotesovec, Feb 04 2014

Crossrefs

Cf. A081919, A081920, A081921.

Programs

  • Mathematica
    With[{nn=20},CoefficientList[Series[Exp[4x]/Sqrt[1-x^2] ,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, May 07 2012 *)

Formula

E.g.f.: exp(4x)/sqrt(1-x^2).
D-finite with recurrence: -a(n) + 4*a(n-1) + (n-1)^2*a(n-2) - 4*(n-1)*(n-2)*a(n-3) = 0. - R. J. Mathar, Nov 09 2012
a(n) ~ n^n * (exp(4) + (-1)^n*exp(-4)) / exp(n). - Vaclav Kotesovec, Feb 04 2014

A246256 Triangular array read by rows. Row n lists the coefficients of the closed form of hypergeometric([1/2, -n/2, (1-n)/2], [], 4*z).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 9, 6, 1, 45, 10, 1, 225, 135, 15, 1, 1575, 315, 21, 1, 11025, 6300, 630, 28, 1, 99225, 18900, 1134, 36, 1, 893025, 496125, 47250, 1890, 45, 1, 9823275, 1819125, 103950, 2970, 55, 1, 108056025, 58939650, 5457375, 207900, 4455, 66, 1
Offset: 0

Views

Author

Peter Luschny, Aug 21 2014

Keywords

Examples

			Triangle starts:
[ 0] 1,
[ 1] 1,
[ 2] 1, 1,
[ 3] 3, 1,
[ 4] 9, 6, 1,
[ 5] 45, 10, 1,
[ 6] 225, 135, 15, 1,
[ 7] 1575, 315, 21, 1,
[ 8] 11025, 6300, 630, 28, 1,
[ 9] 99225, 18900, 1134, 36, 1,
[10] 893025, 496125, 47250, 1890, 45, 1,
[11] 9823275, 1819125, 103950, 2970, 55, 1,
...

Crossrefs

Cf. A081919 (row sums), A138022, A246257.

Programs

  • Maple
    g := exp(x*z)/sqrt((1-z)/(1+z)); gser := n -> series(g, z, n+2):
seq(seq(coeff(n!*coeff(gser(n),z,n),x,2*i+irem(n,2)),i=0..iquo(n,2)),n=0..12);
# Recurrence for A138022 from Robert Israel.
T := proc(n, k) option remember;
if k < 0 or n < k then 0 elif k = n then 1 elif k = n-1 then n
elif k = 0 then T(n-1,k)+(n-2)*(n-1)*T(n-2,k)
else T(n-1,k)+T(n-1,k-1)+(n-2)*(n-1)*(T(n-2,k)-T(n-3,k-1)) fi end:
A246256_row := n -> seq(T(n,2*k+(n mod 2)),k=0..iquo(n,2)):
seq(A246256_row(n), n=0..12);
  • Mathematica
    row[n_] := HypergeometricPFQ[{1/2, -n/2, (1-n)/2}, {}, 4z] // FunctionExpand // CoefficientList[#, z]& // Reverse;
Table[row[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Aug 02 2019 *)
T[n_, k_] := Product[(2*j - 1)^2, {j, 0, Floor[n/2] - k}]*Binomial[n, 2*k + Mod[n,2]]; Flatten[Table[T[n,k],{n, 0, 12},{k, 0 ,Floor[n/2]}]] (*  Detlef Meya, May 05 2024 *)
  • Sage
    from sage.functions.hypergeometric import closed_form
def A246256_row(n):
    R. = ZZ[]
    h = hypergeometric([1/2,-n/2,(1-n)/2], [], 4*z)
    T = R(closed_form(h)).coefficients()
    return T[::-1]
for n in range(13): A246256_row(n)

Formula

For the e.g.f. and a recurrence see the Maple program.
T(n, k) = (Product_{j=0..(floor(n/2) - k)} (2*j - 1)^2)*binomial(n, 2*k + (n mod 2)). - Detlef Meya, May 05 2024
Showing 1-8 of 8 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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