A056040 -id:A056040 - cp's OEIS frontend

A194589 a(n) = A194588(n) - A005043(n); complementary Riordan numbers.

0, 0, 1, 1, 5, 11, 34, 92, 265, 751, 2156, 6194, 17874, 51702, 149941, 435749, 1268761, 3700391, 10808548, 31613474, 92577784, 271407896, 796484503, 2339561795, 6877992334, 20236257626, 59581937299, 175546527727, 517538571125, 1526679067331, 4505996000730

Offset: 0

Peter Luschny, Aug 30 2011

nonn

The inverse binomial transform of a(n) is A194590(n).

Peter Luschny, The lost Catalan numbers.

Cf. A005043, A189912, A194588, A100071, A005717.

# First method, describes the derivation:
A056040 := n -> n!/iquo(n,2)!^2:
A057977 := n -> A056040(n)/(iquo(n,2)+1);
A001006 := n -> add(binomial(n,k)*A057977(k)*irem(k+1,2),k=0..n):
A005043 := n -> `if`(n=0,1,A001006(n-1)-A005043(n-1)):
A189912 := n -> add(binomial(n,k)*A057977(k),k=0..n):
A194588 := n -> `if`(n=0,1,A189912(n-1)-A194588(n-1)):
A194589 := n -> A194588(n)-A005043(n):
# Second method, more efficient:
A100071 := n -> A056040(n)*(n/2)^(n-1 mod 2):
A194589 := proc(n) local k;
(n mod 2)+(1/2)*add((-1)^k*binomial(n,k)*A100071(k+1),k=1..n) end:
# Alternatively:
a := n -> `if`(n<3,iquo(n,2),hypergeom([1-n/2,-n,3/2-n/2],[1,2-n],4)): seq(simplify(a(n)), n=0..30); # Peter Luschny, Mar 07 2017

sf[n_] := With[{f = Floor[n/2]}, Pochhammer[f+1, n-f]/f!]; a[n_] := Mod[n, 2] + (1/2)*Sum[(-1)^k*Binomial[n, k]*2^-Mod[k, 2]*(k+1)^Mod[k, 2]*sf[k+1], {k, 1, n}]; Table[a[n], {n, 0, 10}] (* Jean-François Alcover, Jul 30 2013, from 2nd method *)
Table[If[n < 3, Quotient[n, 2], HypergeometricPFQ[{1 - n/2, -n, 3/2 - n/2}, {1, 2-n}, 4]], {n,0,30}] (* Peter Luschny, Mar 07 2017 *)

a(n):=sum(binomial(n+2,k)*binomial(n-k,k),k,0,(n)/2); /* Vladimir Kruchinin, Sep 28 2015 */

a(n) = sum(k=0, n/2, binomial(n+2,k)*binomial(n-k,k));
vector(30, n, a(n-3)) \\ Altug Alkan, Sep 28 2015

a(n) = sum_{k=0..n} C(n,k)*A194590(k).
a(n) = (n mod 2)+(1/2)*sum_{k=1..n} (-1)^k*C(n,k)*(k+1)$*((k+1)/2)^(k mod 2). Here n$ denotes the swinging factorial A056040(n).
a(n) = PSUMSIGN([0,0,1,2,6,16,45,..] = PSUMSIGN([0,0,A005717]) where PSUMSIGN is from Sloane's "Transformations of integer sequences". - Peter Luschny, Jan 17 2012
A(x) = B'(x)*(1/x^2-1/(B(x)*x)), where B(x)/x is g.f. of A005043. - Vladimir Kruchinin, Sep 28 2015
a(n) = Sum_{k=0..n/2} C(n+2,k)*C(n-k,k). - Vladimir Kruchinin, Sep 28 2015
a(n) = hypergeom([1-n/2,-n,3/2-n/2],[1,2-n],4) for n>=3. - Peter Luschny, Mar 07 2017
a(n) ~ 3^(n + 1/2) / (8*sqrt(Pi*n)). - Vaclav Kotesovec, Feb 17 2024

A219931 Coefficients related to an asymptotic expansion of the logarithm of the central binomial.

Original entry on oeis.org

1, 6, 5, 28, 9, 22, 13, 120, 17, 38, 21, 92, 25, 54, 29, 496, 33, 70, 37, 156, 41, 86, 45, 376, 49, 102, 53, 220, 57, 118, 61, 2016, 65, 134, 69, 284, 73, 150, 77, 632, 81, 166, 85, 348, 89, 182, 93, 1520, 97, 198, 101, 412, 105, 214, 109, 888, 113, 230, 117

Offset: 1

Peter Luschny, Dec 01 2012

An asymptotic expansion of the logarithm of the central binomial (A000984) for n>0 is given by log(binomial(2*n,n)) ~ (n*log(16)-log(Pi)-log(n) + sum_{k>=1}((-4)^(-k)*A002425(k)/a(k)*n^(1-2*k)))/2.

An asymptotic expansion of the logarithm of the swinging factorial (A056040) for n>1 is given by log(swing(n)) ~ (n*log(4)-log(Pi)-(-1)^n*(log(n/2) - (1/2)*sum_{k>=1}((-1)^k*A002425(k)/a(k)*n^(1-2*k))))/2.

			log(binomial(2*n,n)) = n*log(4) - (log(n)+log(Pi))/2 - 1/(8*a(1)*n) + 1/(32*a(2)*n^3) - 1/(128*a(3)*n^5) + 17/(512*a(4)*n^7) - 31/(2048*a(5)*n^9) + 691/(8192*a(6)*n^11) + O(1/n^13).
log(swing(n)) = n*log(2) - (1/2)*log(Pi) - (1/4)*(-1)^n*(2*log(n/2) + 1/(a(1)*n) - 1/(a(2)*n^3) + 1/(a(3)*n^5) - 17/(a(4)*n^7) + 31/(a(5)*n^9) - 691/(a(6)*n^11)) + O(1/n^13).

Peter Luschny, Table of n, a(n) for n = 1..300
Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.

Cf. A006519, A118413, A385054.

Coeff_list := proc(len) local n;
asympt(ln(n/2)/2+lnGAMMA(n/2+1/2)-lnGAMMA(n/2+1),n,2*len+3);
subs(n=1/n,simplify(convert(%,polynom)));
[seq(4*coeff(unapply(%,n)(n),n,2*k+1),k=0..len-1)] end:
A219931_list := n -> denom(Coeff_list(n)); A219931_list(59);

max = 60; s = Normal[Series[Log[x/2]/2+LogGamma[x/2+1/2]-LogGamma[x/2+1], {x, Infinity, 2*max}]] /. x -> 1/x; a[n_] := Denominator[4*Coefficient[s, x^(2*n-1), 1]]; Table[a[n], {n, 1, max}] (* Jean-François Alcover, Feb 17 2014 *)
a[n_] := Denominator[2*EulerE[2*n-1, 1]/(2*n-1)]; Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Apr 04 2014, after Peter Luschny *)

a(n) = denominator(2*E(2*n-1, 1)/(2*n-1)) where E(n, x) is the Euler polynomial. - Peter Luschny, Apr 03 2014

Warning: a(n) != (2*n-1)*2^valuation(n, 2). This was mistakenly assumed several times in the past, see A385054. - Peter Luschny, Jun 17 2025

Edited and incorrect entries removed by Georg Fischer and Peter Luschny, Jun 16 2025

A253665 a(n) = 2^n*n!/(floor(n/2)!)^2.

Original entry on oeis.org

1, 2, 8, 48, 96, 960, 1280, 17920, 17920, 322560, 258048, 5677056, 3784704, 98402304, 56229888, 1686896640, 843448320, 28677242880, 12745441280, 484326768640, 193730707456, 8136689713152, 2958796259328, 136104627929088, 45368209309696, 2268410465484800

Offset: 0

Peter Luschny, Feb 01 2015

nonn

G. C. Greubel, Table of n, a(n) for n = 0..500

Cf. A056040, A253666, A098430.

a := n -> 2^n*n!/iquo(n,2)!^2: seq(a(n), n=0..25);

Table[2^n*n!/Floor[n/2]!^2, {n, 0, 25}] (* Michael De Vlieger, Feb 02 2015 *)
CoefficientList[Series[(1 + 2 (1 - 8 x) x)/(1 - 16 x^2)^(3/2), {x, 0, 20}],x] (* Benedict W. J. Irwin, Aug 15 2016 *)

a(n)=2^n*n!/(n\2)!^2 \\ Charles R Greathouse IV, Aug 25 2016

a(n) = 2^n*A056040(n).
a(2*n) = 4^n*C(2*n, n) = A098430(n).
a(n) = sum(k=0..n, C(n,k)*n!/(floor(n/2)!)^2) = sum(k=0..n, A253666(n,k)).
G.f.: (1+2*(1-8*x)*x)/(1-16*x^2)^(3/2). - Benedict W. J. Irwin, Aug 15 2016

A277393 a(n) = Integral_{x=0..infinity} H_n(x) * exp(-x), where H_n(x) is n-th Hermite polynomial.

Original entry on oeis.org

1, 2, 6, 36, 300, 3000, 35880, 502320, 8038800, 144698400, 2893937760, 63666630720, 1527999802560, 39727994866560, 1112383838966400, 33371515168992000, 1067888485926662400, 36308208521506521600, 1307095506756591552000, 49669629256750478976000

Offset: 0

Vladimir Reshetnikov, Oct 12 2016

nonn

Hermite polynomials can be generalized to non-integer or even complex indexes using their representation as a contour integral (or as a solution to a differential equation), in which case the first formula for a(n) and the functional relation (recurrence) given below remain valid for all complex n.

This is using the "physicist's" version of Hermite polynomials. - Robert Israel, Oct 14 2016

George E. Andrews, Richard Askey, Ranjan Roy, Special Functions, Cambridge University Press (p.278 for Hermite polynomials).

Robert Israel, Table of n, a(n) for n = 0..403
Eric Weisstein's World of Mathematics, Hermite Polynomial, Hermite Differential Equation
Wikipedia, Hermite polynomials

Cf. A001907, A056040, A277472.

a := proc(n) 4^x*sqrt(Pi)*exp(-1/4)*(GAMMA(1+x/2, -1/4)/((-1)^(x/2)*GAMMA((1-x)/2)) + x*GAMMA((x+1)/2, -1/4)/(2*(-1)^((x-1)/2)*GAMMA(1-x/2))); simplify(limit (%,x=n)) end: seq(a(n),n=0..9); # Peter Luschny, Oct 14 2016
a := n -> (cos(Pi*n/2)*GAMMA((n+1)/2)*GAMMA(n/2+1, -1/4) + I*sin(Pi*n/2)*GAMMA(n/2+1)*GAMMA((n+1)/2, -1/4))/(sqrt(Pi)*exp(1/4)*(I/4)^n): seq(a(n), n=0..20); # Vladimir Reshetnikov, Oct 14 2016
f:= n -> int(orthopoly[H](n,t)*exp(-t),t=0..infinity):
map(f, [$0..30]); # Robert Israel, Oct 14 2016

FunctionExpand@Table[4^n Sqrt[Pi] Exp[-1/4] (Gamma[n/2 + 1, -1/4]/((-1)^(n/2) Gamma[(1 - n)/2]) + n  Gamma[(n + 1)/2, -1/4]/(2 (-1)^((n - 1)/2) Gamma[1 - n/2])), {n, 0, 20}]
Table[Integrate[HermiteH[n, x]*Exp[-x], {x, 0, Infinity}], {n, 0, 10}] (* G. C. Greubel, Oct 13 2016 *)
FunctionExpand@Table[2^n*(n!/Floor[n/2]!)*Gamma[Ceiling[(n+1)/2],-1/4]*Exp[-1/4], {n,0,19}] (* Peter Luschny, Oct 17 2016 *)

def A():
    yield 1
    yield 2
    n, a, h, i = 2, 6, -2, 2
    while True:
        yield a
        n += 1
        a *= n << 1
        if is_even(n):
            i += 4
            h *= -i
            a += h
H = A(); print([next(H) for  in range(20)]) # _Peter Luschny, Oct 16 2016

a(n) = 4^n*sqrt(Pi)*exp(-1/4)*(Gamma(1+n/2, -1/4)/((-1)^(n/2)*Gamma((1-n)/2)) + n*Gamma((n+1)/2, -1/4)/(2*(-1)^((n-1)/2)*Gamma(1-n/2))), assuming that 1/Gamma(z) is an entire function of z having zeros at nonpositive integer arguments.
Recurrence: 2*((n+1)*a(n) + 2*n*(n-1)*a(n-2)) = 2*n*a(n-1) + a(n+1).
E.g.f.: exp(-x^2)/(1-2*x).
a(n)/n! ~ exp(-1/4) * 2^n. - Vaclav Kotesovec, Oct 14 2016
a(2*n) = 2^n*(2*n-1)!!*A001907(n), a(2*n+1) = 2^(n+1)*(2*n+1)!!*A001907(n). - Vladimir Reshetnikov, Oct 14 2016
From Peter Luschny, Oct 17 2016: (Start)
a(n) = 2^n*(n!/floor(n/2)!)*Gamma(ceiling((n+1)/2),-1/4)*exp(-1/4).
The swinging factorial A056040(n) divides a(n).
Recurrence: If n is odd then a(n) = a(n-1)*n*2 else a(n) = a(n-1)*n*2 + (-1)^[n/2]* n!/[n/2]!. See the Sage implementation. (End)

A328000 a(n) = Sum_{k=0..n}(k!(n - k)!)/(floor(k/2)!floor((n - k)/2)!)^2.

Original entry on oeis.org

1, 2, 5, 16, 28, 96, 160, 512, 896, 2560, 4864, 12288, 25600, 57344, 131072, 262144, 655360, 1179648, 3211264, 5242880, 15466496, 23068672, 73400320, 100663296, 343932928, 436207616, 1593835520, 1879048192, 7314866176, 8053063680, 33285996544, 34359738368

Offset: 0

Peter Luschny, Oct 01 2019

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,12,0,-48,0,64).

Cf. A327999, A328001.

Cf. A056040, A002699, A001477, A081908, A145018.

[IsOdd(n) select 2^(n - 1)*(n + 1) else 2^(n - 5)*(n*(n + 2) + 32):n in [0..30]]; // Marius A. Burtea, Feb 05 2020

swing := n -> n!/iquo(n,2)!^2: a := n -> add(swing(k)*swing(n-k), k=0..n):
seq(`if`(irem(n, 2) = 0, 2 + n*(n + 2)/16, n + 1)*2^(n - 1), n=0..31);

A328000List[len_] := CoefficientList[Series[(4 x^2 - x - 1)^2 / (1 - 4 x^2)^3 , {x, 0, len}], x]; A328000List[31]
LinearRecurrence[{0,12,0,-48,0,64},{1,2,5,16,28,96},40] (* Harvey P. Dale, Jun 19 2022 *)

x='x + O('x^32);
Vec(serlaplace(((3*x + 8)*sinh(2*x) + (2*x^2 + 16*(x + 1))*cosh(2*x))/16))

Vec((1 + x - 4*x^2)^2 / ((1 - 2*x)^3*(1 + 2*x)^3) + O(x^30)) \\ Colin Barker, Feb 05 2020

a(n) = Sum_{k=0..n} s(k)*s(n-k) where s(n) = A056040(n).
a(n) = [x^n] (4*x^2 - x - 1)^2 / (1 - 4*x^2)^3.
a(n) = 2^(n - 5)*(n*(n + 2) + 32) if n even else 2^(n - 1)*(n + 1).
a(2*n) = A327999(n).
a(2*n-1) = A002699(n), (with a(-1) = 0).
a(2^n-1) = 2^(2^n - 2 + n) for n >= 1.
2*a(2*n)/2^n = A081908(n+1).
4*a(2*n)/4^n = A145018(n+1).
2*a(2*n-1)/4^n = A001477(n).
From Stefano Spezia, Oct 19 2019: (Start)
a(n) = n! [x^n] (1/32)*exp(-2*x)*(8 + exp(4*x)*(8 + x)*(3 + 2*x) + x*(13 + 2*x)).
a(n) = 12*a(n-2) - 48*a(n-4) + 64*a(n-6) for n > 5. (End)

A069945 Let M_k be the k X k matrix M_k(i,j)=1/binomial(i+n,j); then a(n)=1/det(M_(n+1)).

Original entry on oeis.org

1, -6, -360, 252000, 2222640000, -258768639360000, -410299414270986240000, 9061429740221589431500800000, 2835046804394206618956825845760000000, -12733381268715468286016211650968992153600000000

Offset: 1

Benoit Cloitre, Apr 27 2002

If k>n+1 det(M_k)=0

A005249, A056040.

a[n_] := (-1)^Quotient[n, 2]/(Det[HilbertMatrix[n]] n!); Array[a, 10] (* Jean-François Alcover, Jul 06 2019 *)

for(n=0,10,print1(1/matdet(matrix(n+1,n+1,i,j,1/binomial(i+n,j))),","))

def A069945(n): return (-1)^(n//2)*mul(binomial(i,i//2) for i in (1..2*n-1))
[A069945(i) for i in (1..11)] # Peter Luschny, Sep 18 2012

|a(n)| = det(M^(-1)), where M is an n X n matrix with M[i, j]=i/(i+j-1) (or j/(i+j-1)). |a(n)| = 1/det(HilbertMatrix(n))/n! = A005249(n)/n!. - Vladeta Jovovic, Jul 26 2003
|a(n)| = Product_{i=1..2n-1} binomial(i,floor(i/2)). - Peter Luschny, Sep 18 2012
|a(n)| = (Product_{i=1..2n-1} A056040(i))/n! = A163085(2*n-1)/n!. - Peter Luschny, Sep 18 2012

A151332 Number of walks within N^2 (the first quadrant of Z^2) starting and ending at (0,0) and consisting of 4 n steps taken from {(-1, -1), (-1, 1), (1, 0)}.

Original entry on oeis.org

1, 2, 28, 660, 20020, 705432, 27457584, 1147334760, 50561468100, 2322279359400, 110250966574320, 5377893986141040, 268315541493159888, 13645106597301720800, 705378072079232798400, 36985702814877062972880, 1963555139681260758978660, 105393959626252993455319560

Offset: 0

Manuel Kauers, Nov 18 2008

a(n) is also the number of words of 4n length consisting of 2n X's, n Y's and n Z's such that any initial segment of the string has at least as many X's as Y+Z's, and at least as many Y's as Z's. - Istvan Marosi, Apr 27 2014

Alois P. Heinz, Table of n, a(n) for n = 0..558
M. Bousquet-Mélou and M. Mishna, Walks with small steps in the quarter plane, arXiv:0810.4387 [math.CO], 2008.

a:= proc(n) option remember; `if`(n=0, 1,
      (4*(4*n-3)*(2*n-1)*(4*n-1)*a(n-1))/(n*(n+1)*(2*n+1)))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Apr 27 2014
S := proc(a) global x; series(a,x=0,20) end:
ogf := S(int(S(x^(-1/2)*hypergeom([1/4,3/4],[2],64*x)),x)/(2*x^(1/2)));  # Mark van Hoeij, Aug 14 2014

aux[i_Integer, j_Integer, n_Integer] := Which[Min[i, j, n] < 0 || Max[i, j] > n, 0, n == 0, KroneckerDelta[i, j, n], True, aux[i, j, n] = aux[-1 + i, j, -1 + n] + aux[1 + i, -1 + j, -1 + n] + aux[1 + i, 1 + j, -1 + n]]; Table[aux[0, 0, 4 n], {n, 0, 25}]

a(n) = A000108(n)*A000108(2n). - Istvan Marosi, Apr 27 2014
a(n) = A056040(4*n)*A056040(2*n)/A000384(n+1). - Peter Luschny, Apr 28 2014
G.f.: hypergeom([1/4, 1/2, 3/4], [3/2, 2], 64*x). - Robert Israel, Aug 14 2014
D-finite with recurrence n*(n+1)*(2*n+1)*a(n) -4*(4*n-3)*(2*n-1)*(4*n-1)*a(n-1)=0. - R. J. Mathar, Jul 27 2022

A163086 Product of first n terms of A163085.

Original entry on oeis.org

1, 1, 2, 24, 1728, 3732480, 161243136000, 975198486528000000, 412860031256494080000000000, 110116706384632080236544000000000000000, 7401233839469056679744633202278400000000000000000000

Offset: 0

Peter Luschny, Jul 21 2009

nonn

Cf. A056040, A163085, A055462, A000178.

a := proc(n) local i; mul(A163085(i),i=0..n) end;

b[0] = 1; b[n_] := b[n] = b[n-1] n! / Floor[n/2]!^2;
a[n_] := Product[b[k], {k, 0, n}];
Table[a[n], {n, 0, 10}] (* Jean-François Alcover, Jul 11 2019 *)

def A163086(n):
    swing = lambda n: factorial(n)/factorial(n//2)^2
    return mul(swing(i+1)^(n-i) for i in (0..n))
[A163086(i) for i in (0..10)] # Peter Luschny, Sep 18 2012

a(n) = product_{i=0..n} A056040(i+1)^(n-i). - Peter Luschny, Sep 18 2012

A163773 Row sums of the swinging derangement triangle (A163770).

Original entry on oeis.org

1, 1, 4, 15, -14, 185, -454, 2107, -6194, 22689, -70058, 234971, -734304, 2368379, -7404318, 23417955, -72988938, 228324569, -708982738, 2202742447, -6815736144, 21077285943, -65016664062, 200371842727, -616463969324, 1894794918275, -5816606133674, 17839764136377

Offset: 0

Peter Luschny, Aug 05 2009

sign

G. C. Greubel, Table of n, a(n) for n = 0..1000
Peter Luschny, Swinging Factorial.

Cf. A163770.

swing := proc(n) option remember; if n = 0 then 1 elif
irem(n, 2) = 1 then swing(n-1)*n else 4*swing(n-1)/n fi end:
a := proc(n) local i,k; add(add((-1)^(n-i)*binomial(n-k,n-i)*swing(i),i=k..n), k=0..n) end:

sf[n_] := n!/Quotient[n, 2]!^2; t[n_, k_] := Sum[(-1)^(n - i)*Binomial[n - k, n - i]*sf[i], {i, k, n}]; Table[Sum[t[n, k], {k, 0, n}], {n, 0, 50}] (* G. C. Greubel, Aug 03 2017 *)

a(n) = Sum_{k=0..n} Sum_{i=k..n} (-1)^(n-i)*binomial(n-k,n-i)*i$ where i$ denotes the swinging factorial of i (A056040).

Terms a(18) onward added by G. C. Greubel, Aug 03 2017

A163775 Row sums of triangle A163772.

Original entry on oeis.org

1, 11, 73, 403, 2021, 9567, 43611, 193683, 844213, 3629083, 15437951, 65143503, 273148279, 1139548469, 4734740493, 19606960755, 80969809797, 333601494651

Offset: 0

Peter Luschny, Aug 05 2009

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000
Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.
Peter Luschny, Swinging Factorial.

Cf. A163772.

swing := proc(n) option remember; if n = 0 then 1 elif
irem(n, 2) = 1 then swing(n-1)*n else 4*swing(n-1)/n fi end:
a := proc(n) local i,k; add(add((-1)^(n-i)*binomial(n-k,n-i)*swing(2*i+1),i=k..n),k=0..n) end:

sf[n_] := n!/Quotient[n, 2]!^2; t[n_, k_] := Sum[(-1)^(n - i)* Binomial[n - k, n - i]*sf[2*i + 1], {i, k, n}]; Table[Sum[t[n, k], {k, 0, n}], {n, 0, 50}] (* G. C. Greubel, Aug 04 2017 *)

a(n) = Sum_{k=0..n} Sum_{i=k..n} (-1)^(n-i)*binomial(n-k,n-i)*(2i+1)$ where i$ denotes the swinging factorial of i (A056040).

cp's OEIS Frontend

A194589 a(n) = A194588(n) - A005043(n); complementary Riordan numbers.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Maxima

PARI

Formula

A219931 Coefficients related to an asymptotic expansion of the logarithm of the central binomial.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A253665 a(n) = 2^n*n!/(floor(n/2)!)^2.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A277393 a(n) = Integral_{x=0..infinity} H_n(x) * exp(-x), where H_n(x) is n-th Hermite polynomial.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Mathematica

Sage

Formula

A328000 a(n) = Sum_{k=0..n}(k!*(n - k)!)/(floor(k/2)!*floor((n - k)/2)!)^2.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

PARI

Formula

A069945 Let M_k be the k X k matrix M_k(i,j)=1/binomial(i+n,j); then a(n)=1/det(M_(n+1)).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

PARI

Sage

Formula

A151332 Number of walks within N^2 (the first quadrant of Z^2) starting and ending at (0,0) and consisting of 4 n steps taken from {(-1, -1), (-1, 1), (1, 0)}.

Views

Author

Keywords

Comments

A328000 a(n) = Sum_{k=0..n}(k!(n - k)!)/(floor(k/2)!floor((n - k)/2)!)^2.