Groux Roland - cp's OEIS frontend

A188144 Binomial transform A140456(n+1) (indecomposable involutions).

1, 2, 6, 20, 74, 292, 1218, 5308, 24034, 112484, 542346, 2686268, 13639226, 70863652, 376208706, 2038335580, 11259474754, 63353211332, 362819139978, 2113410084188, 12513610048154, 75274067489284, 459782361574146, 2850369932150908, 17926893505949986, 114337654086861092

Offset: 0

Groux Roland, Mar 22 2011

nonn

a(n) is also the INVERTi transform of A005425(n+1) (self-inverse partial permutations) starting at n=2.

Andrew Howroyd, Table of n, a(n) for n = 0..200

Cf. A005425, A140456.

b:= proc(n) b(n):= `if`(n<2, n+1,  2*b(n-1) + (n-1)*b(n-2)) end:
g:= proc(n) g(n):= `if`(n<1, -1, -add(g(n-i)*b(i), i=1..n)) end:
a:= n-> g(n+2):
seq(a(n), n=0..28);  # Alois P. Heinz, Mar 19 2020

nmax = 18; A140456 = CoefficientList[ Series[1 - 1/Total[ CoefficientList[ Series[Exp[x^2/2 + x], {x, 0, nmax + 2}], x]*Range[0, nmax + 2]!* x^Range[0, nmax + 2]], {x, 0, nmax + 2}], x]; a[n_] := Sum[ Binomial[n, k]*A140456[[k + 3]], {k, 0, n}]; Table[a[n], {n, 0, nmax}] (* Jean-François Alcover, Jul 03 2013 *)

seq(n)={Vec(1 - 2*x - 1/serlaplace(exp( 2*x + x^2/2 + O(x^3*x^n) )))} \\ Andrew Howroyd, Jan 06 2020

a(n) is the moment of order n for the probability density function: sqrt(2/Pi^3)*exp((x-2)^2/2)/(1+(erfi((x-2)/sqrt(2)))^2) over the interval -infinity..infinity, with erfi the imaginary error function.

G.f.: A(x) = (1 - 2*x - G(0))/x^2; G(k) = 1 - 2*x - x^2*(k+1)/G(k+1); (continued fraction, 1-step). - Sergei N. Gladkovskii, Jan 26 2012

Terms a(19) and beyond from Andrew Howroyd, Jan 06 2020

A188287 Convolution of A000085 with itself.

Original entry on oeis.org

1, 2, 5, 12, 32, 88, 260, 800, 2604, 8824, 31340, 115568, 443760, 1763456, 7260256, 30835712, 135124496, 609027360, 2822461648, 13417923008, 65401203584, 326242088064, 1664539966400, 8674167861760, 46140838036160, 250248380068736, 1383064482739392, 7782094359642880

Offset: 0

Groux Roland, Mar 26 2011

nonn

a(n) is also the moment of order n for the measure of density: x*exp(-(x-1)^2)*erfi((x-1)/sqrt(2)) over the interval -infinity..infinity, with erfi the Imaginary Error Function.

Andrew Howroyd, Table of n, a(n) for n = 0..200

Cf. A000085, A034430.

seq(n)={Vec(serlaplace(exp(x + x^2/2 + O(x*x^n)))^2)} \\ Andrew Howroyd, Nov 04 2019

a(n) = Sum_{k=0..floor(n/2)} binomial(n+1,2*k+1)*A034430(k).

Terms a(20) and beyond from Andrew Howroyd, Nov 04 2019

A188143 Binomial transform of A187848.

Original entry on oeis.org

1, 5, 29, 193, 1453, 12209, 113237, 1149241, 12675661, 151095569, 1937411429, 26614052617, 390244490749, 6087782363009, 100728768290645, 1762767028074937, 32542231109506285, 632202858036492593, 12895661952702667205

Offset: 0

Groux Roland, Mar 24 2011

nonn

a(n) is also the INVERTi transform of A010842(n+1) starting at n=2.
a(n) is also the moment of order n for the measure of density: exp(x-2) / ((Ei(x-2))^2+Pi^2) over the interval 2..infinity with Ei the exponential integral.
More generally, for every integer k, the sequence a(n,k)=int(x^n*exp(x-k) / ((Ei(x-k))^2+Pi^2), x=k..infinity) is the INVERTi transform of the sequence b(n+1,k), starting at n=2, with b(n,k)=int(x^n*exp(x-k), x=k..infinity) whose e.g.f. is exp(k*x)/(1-x).

Vaclav Kotesovec, Table of n, a(n) for n = 0..200

Cf. A000023.

with(LinearAlgebra):
c:= proc(n) option remember; add(n!/k!, k=0..n) end:
b:= n-> (-1)^(n+1) * Determinant(Matrix(n+2,
        (i, j)-> `if`(0<=i+1-j, c(i+1-j), 0))):
a:= proc(n) add(b(k) *binomial(n,k), k=0..n) end:
seq(a(n), n=0..20);  # Alois P. Heinz, Mar 24 2011

c[n_] := c[n] = Sum[n!/k!, {k, 0, n}]; b[n_] := (-1)^(n+1)*Det[Table[If[0 <= i+1-j, c[i+1-j], 0], {i, 1, n+2}, {j, 1, n+2}]]; a[n_] := Sum[b[k] * Binomial[n, k], {k, 0, n}]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Feb 28 2017, after Alois P. Heinz *)

a(n) = Integral_{x=2..oo} x^n*exp(x-2)/((Ei(x-2))^2 + Pi^2) dx.
G.f.: 1/x^2 - 3/x - Q(0)/x^2, where Q(k) = 1 - 2*x - x*(k+1)/(1-x*(k+1)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, Apr 19 2013
a(n) ~ exp(2) * n^2 * n!. - Vaclav Kotesovec, Nov 02 2023

A187870 Numerator of the coefficient of x^(2n) in the expansion of 1/x^4 - 1/(3x^2) - 1/(x^3arctanh(x)).

Original entry on oeis.org

4, 44, 428, 10196, 10719068, 25865068, 5472607916, 74185965772, 264698472181028, 2290048394728148, 19435959308462817284, 2753151578548809148, 20586893910854623222436, 134344844535611780572028924

Offset: 0

Groux Roland, Mar 14 2011

Let f(x) = 1/x^4 - 1/(3*x^2) - 1/(x^3*arctanh(x)) = Sum_{n>=0} r(n)*x^(2n), then a(n) is the numerator of r(n), and r(n) is also the moment of order n for the density rho(x) = 2*sqrt(x)/(4*(arctanh(sqrt(x)))^2 + Pi^2) over the interval [0,1].
r(n) can also be evaluated as (-1)^(n+1)*det(An) with An the square matrix of order n+2 defined by: if j <= i A[i,j] = 1/(2*i-2*j+3), A[i,i+1]=1, if j > i+1 A[i,j]=0.
A very similar sequence of numerators 1, 1, 4, 44, 428, 10196, ... (from there on apparently the same as here) is constructed from the fractions c(0)=-1 and c(n) = Sum_{i=0..n-1} c(i)/(2n-2i+1), which is c(0)=-1, c(1)=1/3, c(2)=4/45, c(3)= 44/945, etc. The recurrence is designed to ensure that Sum_{i=0..n} c(i)/(2n-2i+1) = 0. - Paul Curtz, Sep 15 2011
Prepending 1 to the data gives the (-1)^n times the numerator of the odd powers in the expansion of 1/arctan(x). - Peter Luschny, Oct 04 2014

Cf. A195466 (denominator).

A187870 := proc(n)
        1/x^4 -1/(3*x^2) -1/(x^3*arctanh(x)) ;
        coeftayl(%,x=0,2*n) ;
        numer(%) ;
end proc:
seq(A187870(n),n=0..10) ; # R. J. Mathar, Sep 21 2011
# Or
seq((-1)^n*numer(coeff(series(1/arctan(x),x,2*n+2),x,2*n+1)),n=1..14); # Peter Luschny, Oct 04 2014

a[n_] := Sum[(2^(j+1)*Binomial[2*n+3, j]*Sum[(k!*StirlingS1[j+k, j]*StirlingS2[j+1, k])/(j+k)!, {k, 0, j+1}])/(j+1), {j, 0, 2*n+3}]/ (2*n+3); Table[a[n] // Numerator, {n, 0, 13}] (* Jean-François Alcover, Jul 03 2013, after Vladimir Kruchinin's formula in A216272 *)

A187848 a(n) is the moment of order n for the probability density function defined by rho(x)=exp(x-1)/((Ei(x-1))^2+Pi^2) over the interval 1..infinity, with Ei the exponential integral.

Original entry on oeis.org

1, 4, 20, 120, 836, 6608, 58324, 568296, 6060340, 70245856, 879937892, 11853424536, 170963881892, 2629912684784, 42995842035316, 744683072665416, 13624184625098644, 262594854417561856, 5319099368762699012, 112977659152942035192, 2511041582408699358980

Offset: 0

Groux Roland, Mar 14 2011

nonn

a(n) is also the binomial transform of A003319(n+1).

Alois P. Heinz, Table of n, a(n) for n = 0..200

with(LinearAlgebra):
c:= proc(n) option remember; add(n!/k!, k=0..n) end:
a:= n-> (-1)^(n+1) *Determinant(Matrix(n+2,
        (i, j)-> `if`(0<=i+1-j, c(i+1-j), 0))):
seq(a(n), n=0..20);  # Alois P. Heinz, Mar 24 2011
# second Maple program:
b:= proc(n) option remember;
      `if`(n<0, -1, -add(b(n-i)*i!, i=1..n+1))
    end:
a:= n-> add(b(k+1)*binomial(n, k), k=0..n):
seq(a(n), n=0..20);  # Alois P. Heinz, Aug 26 2013

b[n_] := b[n] = If[n<0, -1, -Sum[b[n-i]*i!, {i, 1, n+1}]]; a[n_] := Sum[b[k+1] * Binomial[n, k], {k, 0, n}]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jun 30 2015, after Alois P. Heinz *)

Let c(n)=A000522(n) and An the square matrix of order n+2 defined by: if j<=i A[i,j]=c(i-j+1); A(i,i+1)=1; if j>i+1 A[i,j]=0; then a(n)=(-1)^(n+1)*det(An).
G.f.: (1 - 2*x - U(0))/x^2 where U(k)=  1 - x - x*(k+1)/(1 - x*(k+1)/U(k+1)) ; (continued fraction, 2-step). - Sergei N. Gladkovskii, Oct 14 2012
G.f.: 1/x^2 - 1/x  -1/(x^2*W(0)) , where W(k) = 1 - x*(k+1)/( x*(k+1) - 1/(1 - x*(k+1)/( x*(k+2) - 1/W(k+1) ))); (continued fraction). - Sergei N. Gladkovskii, Aug 25 2013
G.f.: W(0)/x - 1/x, where W(k) = 1 - x*(k+1)/( x*(k+2) - 1/(1 - x*(k+1+R)/( x*(k+1+R) - 1/W(k+1) ))); R=1 (continued fraction). - Sergei N. Gladkovskii, Aug 26 2013
a(n) ~ exp(1) * n! * n^2 * (1 - 1/n - 4/n^3 - 23/n^4 - 175/n^5 - 1615/n^6 - 17375/n^7 - 212607/n^8 - 2909007/n^9 - 43953071/n^10). - Vaclav Kotesovec, Sep 02 2014, updated Aug 01 2015

A180255 a(n) = n^2 * a(n-1) + n, a(0)=0.

Original entry on oeis.org

0, 1, 6, 57, 916, 22905, 824586, 40404721, 2585902152, 209458074321, 20945807432110, 2534442699285321, 364959748697086236, 61678197529807573897, 12088926715842284483826, 2720008511064514008860865, 696322178832515586268381456, 201237109682597004431562240801

Offset: 0

Groux Roland, Jan 17 2011

nonn

Integral_{x=0..1} x^n*BesselI(0,2*x^(1/2)) dx = A006040(n)*BesselI(1,2) - a(n)*BesselI(0,2). An elementary consequence is the irrationality of BesselI(0,2)/BesselI(1,2).

Paolo Xausa, Table of n, a(n) for n = 0..250 (terms 0..41 from Vincenzo Librandi)

Cf. A006040, A007526, A066998, A228229.

FoldList[#2^2*# + #2 &, Range[0, 20]] (* Paolo Xausa, Jun 19 2025 *)

a[0]:0$ a[n]:=n^2*a[n-1]+n$ makelist(a[n], n, 0, 15); /* Bruno Berselli, May 23 2011 */

a(n)=if(n==0,0,(n)^2*a(n-1)+(n));
for(n=0,12,print1(a(n),", "));  /* show terms */

From Seiichi Manyama, Jan 05 2024: (Start)
a(n) = (n!)^2 * Sum_{k=0..n} k/(k!)^2.
a(n) = n * A228229(n-1) for n > 0. (End)

A178172 Numerators of coefficients in 1/(log(1+x)*log(1-x)) - 5/12 + 1/x^2 power series.

Original entry on oeis.org

7, 2699, 103669, 1088429, 40938135737, 78638210671, 294591811249, 89540991316447487, 72003397880992782229, 254817218437479631303, 297853453602437702393397023, 3199628601055396119933058397, 558554606804743021307746074767

Offset: 1

Groux Roland, Dec 17 2010

nonn

a(n) is the numerator of the moment of order n for the density 1/((log(1 + 1/sqrt(x))*(log^2(1/sqrt(x) - 1) + Pi^2)) over the interval 0..1.

Rest@ Union@ Numerator@ CoefficientList[ Series[1/(Log[1 + x]*Log[1 - x]) - 5/12 + 1/x^2, {x, 0, 26}], x]

A177354 a(n) is the moment of order n for the density measure 24x^4exp(-x)/( (x^4exp(-x)Ei(x) - x^3 - x^2 - 2x - 6)^2 + Pi^2x^8exp(-2x) ) over the interval 0..infinity.

Original entry on oeis.org

5, 35, 305, 3095, 35225, 439775, 5939225, 85961375, 1324702025, 21632195375, 372965377625, 6769644905375, 129049505347625, 2578419996023375, 53898389265685625, 1176832196718869375, 26798832693476455625, 635575680349115699375, 15677971277701873945625, 401729457433222058609375

Offset: 0

Groux Roland, Dec 10 2010

nonn

Ei(.) is the exponential integral.
This is the case k=4 in the family a(n,k) = (1/k!)*( (n+k+2)!-(k+1)*(n+k+1)! -Sum_{i=0..n-1} (n+k-i)!*a(i,k) ). The values k = 0 to 3 are represented by A003319, A111537, A111546, and A111556.
a(n,k) is the moment of order n for the density k!*x^k*exp(-x)/((x^k*exp(-x)*Ei(x) - Pk(x))^2 + Pi^2*x^(2*k)*exp(-2*x)) on the interval 0..infinity with polynomials Pk(x) = Sum_{i=0..k-1} (k-1-i)!*x^i.

R. Groux, Polynômes orthogonaux et transformations intégrales, Cépadués, 2008, 125-129.

a(n)=if(n==0, 5, (1/24)*( (n+6)! -5*(n+5)! -sum(i=0,n-1, (n+4-i)!*a(i) ) ) ); \\ Joerg Arndt, May 04 2013

a(n) = (1/24)*( (n+6)! - 5*(n+5)! - Sum_{i=0..n-1} (n+4-i)!*a(i) ).
a(n) = 5*A111532(n+1) (conjecture). - R. J. Mathar, Dec 14 2010
G.f.: 1/x/Q(0) - 1/x, where Q(k) = 1 - 3*x + k*x - x*(k+2)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 04 2013
G.f.: (1-x-2/G(0))/x^2, where G(k) = 1 + 1/(1 - x*(k+1)/(x*(k-1) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 05 2013
G.f.: 1/x^2 - 5/x - 2/(x^2*G(0)), where G(k) = 1 + 1/(1 - x*(k+5)/(x*(k+5) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 06 2013

A167872 A sequence of moments connected with Feynman numbers (A000698): Half the number of Feynman diagrams of order 2(n+1), for the electron self-energy in quantum electrodynamics (QED), i.e., all proper diagrams including Furry vanishing diagrams (those that vanish in 4-dimensional QED because of Furry theorem).

Original entry on oeis.org

1, 3, 21, 207, 2529, 36243, 591381, 10786527, 217179009, 4782674403, 114370025301, 2952426526767, 81864375589089, 2427523337157363, 76683680366193621, 2571609710380950207, 91265370849151405569, 3417956847888948899523

Offset: 0

Groux Roland, Nov 14 2009

nonn

a(n) is the moment of order 2*n of the probability density function defined by rho(x) = sqrt(Pi/2)*exp(-x^2/2)/((x*phi(x)+1)^2 + Pi^2*x^2*exp(-x^2)), where phi(x) = Integral_{t=-oo..oo} t*log(abs(x-t))*exp(-t^2/2) dt.

			G.f. = 1 + 3*x + 21*x^2 + 207*x^3 + 2529*x^4 + 36243*x^5 + 591381*x^6 + ...

Roland Groux. Polynômes orthogonaux et transformations intégrales. Cepadues. 2008. pages 195..206.

Alois P. Heinz, Table of n, a(n) for n = 0..400
Trinh Khanh Duy and Tomoyuki Shirai, The mean spectral measures of random Jacobi matrices related to Gaussian beta ensembles, arXiv preprint arXiv:1504.06904 [math.SP], 2015.
Adrian Ocneanu, On the inner structure of a permutation: bicolored partitions and Eulerians, trees and primitives; arXiv preprint arXiv:1304.1263 [math.CO], 2013.
Wikipedia, Feynman diagram

Row 3 of A321964. Cf. A000698, A115974, A005411, A005412, A001147.

(* f = A000698 *) f[n_] := f[n] = (2*n - 1)!! - Sum[f[n - k]*(2*k - 1)!!, {k, 1, n - 1}]; a[n_] := a[n] = f[n + 2]/2 - Sum[f[n + 1 - k]*a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 17}] (* Jean-François Alcover, Jul 03 2013, from 3rd formula *)
nmax = 20; CoefficientList[Series[1/(1 + x + ContinuedFractionK[-(k - (-1)^k)*x, 1, {k, 3, nmax}]), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 06 2022, after Peter Bala *)

{a(n) = local(A); n++; if( n<1, 0, A = vector(n); A[1] = 1; for( k=2, n, A[k] = (2*k - 3) * A[k-1] + 2 * sum( j=1, k-1, A[j] * A[k-j])); A[n])}; /* Michael Somos, Jul 23 2011 */

Sum_{n>=0} a(n)/z^(2n+1) = (1/2)*(z-S(z)/(z*S(z)-1)) with S(z) = Sum_{n>=0} (2*n)!/(2^n*n!*z^(2*n+1)).
a(n) = (2*n - 1) * a(n-1) + 2 * Sum_{k=1..n} a(k-1) * a(n-k) if n>0. - Michael Somos, Jul 23 2011
a(0)=1; for n > 0, a(n) = A000698(n+2)/2 - Sum_{k=0..n-1} A000698(n+1-k)*a(k).
G.f.: 1/(1-3*x/(1-4*x/(1-5*x/(1-6*x/(1-7*x/(1-8*x/(...))))))) (continued fraction). - Philippe Deléham, Nov 20 2011
G.f.: 1/Q(0), where Q(k) = 1 - x*(k+3)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 20 2013
Let A(x) be the g.f. of A127059 and B(x) be the g.f. of A167872. Then A(x) = (1 - 1/B(x))/x.
G.f.: 1/Q(0), where Q(k) = 1 - x*(2*k+3)/(1 - x*(2*k+4)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, May 21 2013
G.f.: G(0)/2, where G(k) = 1 + 1/(1 - (2*k+3)*x/((2*k+2)*x + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 14 2013
G.f.: G(0), where G(k) = 1 - x*(k+3)/(x*(k+3) - 1/G(k+1)); (continued fraction). - Sergei N. Gladkovskii, Aug 05 2013
a(n) = A115974(n)/2, see comments in A115974. See also A000698, A005411, A005412. - Robert Coquereaux, Sep 14 2014
a(n) ~ 2^(n + 3/2) * n^(n+2) / exp(n). - Vaclav Kotesovec, Jan 02 2019
G.f.: 1/(1 + x - 4*x/(1 - 3*x/(1 - 6*x/(1 - 5*x/(1 - 8*x/(1 - 7*x/(1 - ...))))))). - Peter Bala, May 30 2022

Name clarified from Robert Coquereaux, Sep 14 2014

cp's OEIS Frontend

User: Groux Roland

A188144 Binomial transform A140456(n+1) (indecomposable involutions).

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A188287 Convolution of A000085 with itself.

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A188143 Binomial transform of A187848.

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A187870 Numerator of the coefficient of x^(2n) in the expansion of 1/x^4 - 1/(3*x^2) - 1/(x^3*arctanh(x)).

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A187848 a(n) is the moment of order n for the probability density function defined by rho(x)=exp(x-1)/((Ei(x-1))^2+Pi^2) over the interval 1..infinity, with Ei the exponential integral.

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A180255 a(n) = n^2 * a(n-1) + n, a(0)=0.

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A178172 Numerators of coefficients in 1/(log(1+x)*log(1-x)) - 5/12 + 1/x^2 power series.

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A177354 a(n) is the moment of order n for the density measure 24*x^4*exp(-x)/( (x^4*exp(-x)*Ei(x) - x^3 - x^2 - 2*x - 6)^2 + Pi^2*x^8*exp(-2*x) ) over the interval 0..infinity.

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A187870 Numerator of the coefficient of x^(2n) in the expansion of 1/x^4 - 1/(3x^2) - 1/(x^3arctanh(x)).

A177354 a(n) is the moment of order n for the density measure 24x^4exp(-x)/( (x^4exp(-x)Ei(x) - x^3 - x^2 - 2x - 6)^2 + Pi^2x^8exp(-2x) ) over the interval 0..infinity.