A090998 -id:A090998 - cp's OEIS frontend

A163930 Duplicate of A090998.

9, 8, 9, 0, 5, 5, 9, 9, 5, 3, 2, 7, 9, 7, 2, 5, 5, 5, 3, 9, 5, 3, 9, 5, 6, 5, 1, 5, 0, 0, 6, 3, 4, 7, 0, 7, 9, 3, 9, 1, 8, 3, 5, 2, 0, 7, 2, 8, 2, 1, 4, 0, 9, 0, 4, 4, 3, 1, 9, 5, 7, 8, 3, 6, 8, 6, 1, 3, 6, 6, 3, 2, 0, 4, 9, 4, 7, 8, 7, 7, 1, 7, 4, 7, 4, 4, 6, 0, 8, 4, 6, 2, 5, 7, 3, 7, 3, 4, 1, 3, 0, 3, 5, 2

Offset: 0

Johannes W. Meijer and Nico Baken, Aug 13 2009, Aug 17 2009

dead

The higher order exponential integrals, see A163931, are defined by E(x,m,n) = x^(n-1)*int(E(t,m-1,n)/t^n, t=x..infinity) for m>=1 and n>=1, with E(x,m=0,n) = exp(-x).
The series expansions of the higher order exponential integrals are dominated by the gamma(k,n) and the alpha(k,n) constants, see A163927.
The values of the gamma(k,n) = G(k,n) coefficients can be determined with the Maple program.

			G(2,1) = 0.9890559953279725553953956515...

G. C. Greubel, Table of n, a(n) for n = 0..5000
J. W. Meijer and N. H. G. Baken, The Exponential Integral Distribution, Statistics and Probability Letters, Volume 5, No.3, April 1987. pp 209-211.

Cf. A163931 (E(x,m,n)), A163927 (alpha(k,n)).
G(1,1) equals A001620 (gamma).
(gamma - G(1,n)) equals A001008(n-1)/A002805(n-1) for n>=2.
The structure of the G(k,n=1) formulas lead (replace gamma by G and Zeta by Z) to A036039.

ncol:=1; nmax:=5; kmax:=nmax; for n from 1 to nmax do G(0,n):=1 od: for n from 1 to nmax do for k from 1 to kmax do G(k,n):= expand((1/k)*((gamma-sum(p^(-1),p=1..n-1))* G(k-1,n)+sum((Zeta(k-i)-sum(p^(-(k-i)),p=1..n-1))*G(i,n),i=0..k-2))) od; od: for k from 0 to kmax do G(k,ncol):=G(k,ncol) od;

RealDigits[ N[ EulerGamma^2/2 + Pi^2/12, 105]][[1]] (* Jean-François Alcover, Nov 07 2012, from 1st formula *)

G(2,1) = gamma(2,1) = gamma^2/2+Pi^2/12.
G(k,n) = (1/k)*(gamma*G(k-1,n)) - (1/k)*Sum_{p=1..n-1}(p^(-1))* G(k-1,n) + (1/k)* Sum_{i=0..k-2}(Zeta(k-i) * G(i,n))  - (1/k)*Sum_{i=0..k-2}(Sum_{p=1..n-1}(p^(i-k)) * G(i,n)) with G(0,n) = 1 for k>=0 and n>=1.
G(k,n+1) = G(k,n) -G(k-1,n)/n.
GF(z,n) = GAMMA(n-z)/GAMMA(n).

A163931 Decimal expansion of the higher-order exponential integral E(x, m=2, n=1) at x=1.

Original entry on oeis.org

0, 9, 7, 8, 4, 3, 1, 9, 7, 2, 1, 6, 6, 7, 0, 1, 7, 9, 3, 2, 5, 5, 3, 7, 7, 8, 9, 0, 4, 5, 2, 8, 0, 0, 8, 2, 7, 6, 9, 5, 8, 2, 2, 6, 9, 5, 3, 0, 2, 6, 5, 7, 6, 5, 5, 7, 4, 4, 2, 1, 2, 4, 2, 4, 5, 4, 4, 7, 1, 3, 7, 6, 2, 6, 1, 4, 0, 9, 0, 4, 8, 8, 7, 3, 6, 9, 6, 0, 4, 8, 9, 1, 8, 5, 5, 5, 0, 8, 9, 4, 5, 4, 6, 7, 0

Offset: 0

Johannes W. Meijer and Nico Baken, Aug 13 2009, Aug 17 2009

We define the higher-order exponential integrals by E(x,m,n) = x^(n-1)*Integral_{t=x..infinity} E(t,m-1,n)/t^n for m >= 1 and n >= 1 with E(x,m=0,n) = exp(-x), see Meijer and Baken.
The properties of the E(x,m,n) are analogous to those of the well-known exponential integrals E(x,m=1,n), see Abramowitz and Stegun and the formulas.
The series expansions of the higher-order exponential integrals are dominated by the constants alpha(k,n), see A163927, and gamma(k,n) = G(k,n), see A090998.
For information about the asymptotic expansion of the E(x,m,n) see A163932.
Values of E(x,m,n) can be evaluated with the Maple program.

			E(1,2,1) = 0.09784319721667017932553778904528008276958226953026576557442124245....

G. C. Greubel, Table of n, a(n) for n = 0..5000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972, Chapter 5, pp. 227-251.
J. W. Meijer and N. H. G. Baken, The Exponential Integral Distribution, Statistics and Probability Letters, Volume 5, No.3, April 1987. pp 209-211.
M. S. Milgram, The generalized integro-exponential function, Math. of Computation, Vol. 44, pp. 443-458, 1985.
Eric Weisstein's World of Mathematics, The Exponential Integral.

Cf. A163927 (alpha(k,n)), A090998 (gamma(k,n) = G(k,n)), A163932.
Cf. A068985 (E(x=1,m=0,n) = exp(-1)) and A099285 (E(x=1,m=1,n=1)).
Cf. A001563 (n*n!), A002775 (n^2*n!), A091363 (n^3*n!) and A091364 (n^4*n!).

E:= proc(x,m,n) local nmax, kmax, EI, k1, k2, n1, n2; option remember: nmax:=20; kmax:=20; k1:=0: for n1 from 0 to nmax do alpha(k1,n1):=1 od: for k1 from 1 to kmax do for n1 from 1 to nmax do alpha(k1,n1) := (1/k1)*sum(sum(p^(-2*(k1-i1)),p=0..n1-1)*alpha(i1, n1),i1=0..k1-1) od; od: for n2 from 0 to kmax do G(0,n2):=1 od: for n2 from 1 to nmax do for k2 from 1 to kmax do G(k2,n2):=(1/k2)*(((gamma-sum(p^(-1),p=1..n2-1))*G(k2-1,n2)+ sum((Zeta(k2-i2)-sum(p^(-(k2-i2)), p=1..n2-1))*G(i2,n2),i2=0..k2-2))) od; od: EI:= evalf((-1)^m*((-x)^(n-1)/(n-1)!*sum(alpha(kz,n)*(G(m-2*kz,n)+sum(G(m-2*kz-i,n)*ln(x)^i/i!,i=1..m-2*kz)), kz=0..floor(m/2)) + sum((-x)^kx/((kx-n+1)^m*kx!),kx=0..n-2) + sum((-x)^ky/((ky-n+1)^m*ky!),ky=n..infinity))); return(EI): end:

Join[{0}, RealDigits[ N[ EulerGamma^2/2 + Pi^2/12 - HypergeometricPFQ[{1, 1, 1}, {2, 2, 2}, -1], 104]][[1]]] (* Jean-François Alcover, Nov 07 2012, from 1st formula *)

t=1; Euler^2/2 + Pi^2/12 + sumalt(k=1, t*=k; (-1)^k/(k^2*t)) \\ Charles R Greathouse IV, Nov 07 2016

E(x=1,m=2,n=1) = gamma^2/2 + Pi^2/12 + Sum_{k>=1} ((-1)^k/(k^2*k!)).
E(x=0,n,m) = (1/(n-1))^m for n >= 2.
Integral_{t=0..x} E(t,m,n) = 1/n^m - E(x,n,n+1).
dE(x,m,n+1)/dx = - E(x,m,n).
E(x,m,n+1) = (1/n)*(E(x,m-1,n+1) - x*E(x,m,n)).
E(x,m,n) = (-1)^m * ((-x)^(n-1)/(n-1)!) * Sum_{kz=0..floor(m/2)}(alpha (kz, n)*G(m-2*kz, n)) + (-1) ^m * ((-x)^(n-1)/(n-1)!) * Sum_{kz=0..floor(m/2)}(Sum_{i=1..m-2*kz}(alpha (kz, n) *G(m-2*kz-i, n)*log(x)^i/i!)) + (-1)^m * Sum_{ kx=0..n-2}((-x)^kx/((kx-n+1)^m*kx!) + (-1)^m * Sum_{ky>=n}((-x)^ky /(( ky-n+1)^m*ky!)).

A112302 Decimal expansion of quadratic recurrence constant sqrt(1 * sqrt(2 * sqrt(3 * sqrt(4 * ...)))).

Original entry on oeis.org

1, 6, 6, 1, 6, 8, 7, 9, 4, 9, 6, 3, 3, 5, 9, 4, 1, 2, 1, 2, 9, 5, 8, 1, 8, 9, 2, 2, 7, 4, 9, 9, 5, 0, 7, 4, 9, 9, 6, 4, 4, 1, 8, 6, 3, 5, 0, 2, 5, 0, 6, 8, 2, 0, 8, 1, 8, 9, 7, 1, 1, 1, 6, 8, 0, 2, 5, 6, 0, 9, 0, 2, 9, 8, 2, 6, 3, 8, 3, 7, 2, 7, 9, 0, 8, 3, 6, 9, 1, 7, 6, 4, 1, 1, 4, 6, 1, 1, 6, 7, 1, 5, 5, 2, 8

Offset: 1

Michael Somos, Sep 02 2005

From Johannes W. Meijer, Jun 27 2016: (Start)
With Phi(z, p, q) the Lerch transcendent, define LP(n) = (1/n) * sum(Phi(1/2, n-k, 1) * LP(k), k=0..n-1), with LP(0) = 1. Conjecture: Lim_{n -> infinity} LP(n) = A112302.
For similar formulas, see A090998 and A135002. For background information, see A274181.
The structure of the n! * LP(n) formulas leads to the multinomial coefficients A036039. (End)

			1.6616879496335941212958189227499507499644186350250682081897111680...

S. R. Finch, Mathematical Constants, Cambridge University Press, Cambridge, 2003, p. 446.
S. Ramanujan, Collected Papers, Ed. G. H. Hardy et al., AMS Chelsea 2000. See Appendix I. p. 348.

Robert G. Wilson v, Table of n, a(n) for n = 1..1011
Steven Finch, Errata and Addenda to Mathematical Constants, arXiv:2001.00578 [math.HO], 2020, Section 6.10.
Hibiki Gima, Toshiki Matsusaka, Taichi Miyazaki, and Shunta Yara, On integrality and asymptotic behavior of the (k,l)-Göbel sequences, arXiv:2402.09064 [math.NT], 2024. See p. 2.
Olivier Golinelli, Remote control system of a binary tree of switches - II. balancing for a perfect binary tree, arXiv:2405.16968 [cs.DM], 2024. See p. 17.
M. D. Hirschhorn, A note on Somos' quadratic recurrence constant, J. Number Theory 131 (2011), 2061-2063.
Dawei Lu and Zexi Song, Some new continued fraction estimates of the Somos' quadratic recurrence constant, Journal of Number Theory, 155 (2015), 36-45.
Dawei Lu, Xiaoguang Wang, and Ruiqing Xu, Some New Exponential-Function Estimates of the Somos' Quadratic Recurrence Constant, Results in Mathematics 74(1) (2019), Article 6.
Cristinel Mortici, Estimating the Somos' quadratic recurrence constant, J. Number Theory 130 (2010), 2650-1657.
Jesús Guillera and Jonathan Sondow, Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent, arXiv:math/0506319 [math.NT], 2005-2006; see page 8.
Jesús Guillera and Jonathan Sondow, Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent, Ramanujan J. 16 (2008), 247-270.
Jörg Neunhäuserer, On the universality of Somos' constant, arXiv:2006.02882 [math.DS], 2020.
Jonathan Sondow and Petros Hadjicostas, The generalized-Euler-constant function gamma(z) and a generalization of Somos's quadratic recurrence constant, arXiv:math/0610499 [math.CA], 2006.
Jonathan Sondow and Petros Hadjicostas, The generalized-Euler-constant function gamma(z) and a generalization of Somos's quadratic recurrence constant, J. Math. Anal. Appl. 332 (2007), 292-314.
Eric Weisstein's World of Mathematics, Somos's Quadratic Recurrence Constant.
Wikipedia, Somos' quadratic recurrence constant
Xu You and Di-Rong Chen, Improved continued fraction sequence convergent to the Somos' quadratic recurrence constant, Mathematical Analysis and Applications, 436(1) (2016), 513-520.

Cf. A052129, A055209, A116603, A123851, A123852, A123853, A123854.
Cf. A114124 (log).
Cf. A036039, A090998, A135002, A188834, A259235, A274181.

RealDigits[ Fold[ N[ Sqrt[ #2*#1], 128] &, Sqrt@ 351, Reverse@ Range@ 350], 10, 111][[1]] (* Robert G. Wilson v, Nov 05 2010 *)
Exp[-Derivative[1, 0][PolyLog][0, 1/2]] // RealDigits[#, 10, 105]& // First (* Jean-François Alcover, Apr 07 2014, after Jonathan Sondow *)

{a(n) = if( n<-1, 0, n++; default( realprecision, n+2); floor( prodinf( k=1, k^2^-k)* 10^n) % 10)};

prodinf(n=1,n^2^-n) \\ Charles R Greathouse IV, Apr 07 2013

from mpmath import polylog, diff, exp, mp
mp.dps = 120
somos_const = exp(-diff(lambda n: polylog(n, 1/2), 0))
A112302 = [int(d) for d in mp.nstr(somos_const, n=mp.dps)[:-1] if d != '.']  # Jwalin Bhatt, Nov 23 2024

Equals Product_{n>=1} n^(1/2^n). - Jonathan Sondow, Apr 07 2013
Equals exp(A114124) = A188834/2 = sqrt(A259235). - Hugo Pfoertner, Nov 23 2024
From Jwalin Bhatt, Apr 02 2025: (Start)
Equals exp(-PolyLog'(0,1/2)), where PolyLog'(x,y) represents the derivative of the polylogarithm w.r.t. x.
Equals Product_{n>=1} (1+1/n)^(1/2^n).
Equals exp(Sum_{n>=2} log(n)/2^n).
Equals 2*exp(Sum_{n>=1} (log(1+1/n)-1/n)/2^n). (End)

A135002 Decimal expansion of 2/e.

Original entry on oeis.org

7, 3, 5, 7, 5, 8, 8, 8, 2, 3, 4, 2, 8, 8, 4, 6, 4, 3, 1, 9, 1, 0, 4, 7, 5, 4, 0, 3, 2, 2, 9, 2, 1, 7, 3, 4, 8, 9, 1, 6, 2, 2, 2, 6, 2, 0, 6, 3, 5, 3, 5, 6, 6, 9, 0, 1, 5, 6, 7, 3, 6, 0, 3, 3, 9, 4, 9, 2, 2, 9, 9, 1, 4, 8, 9, 7, 9, 9, 6

Offset: 0

Omar E. Pol, Nov 15 2007

From Johannes W. Meijer, Jun 27 2016: (Start)
This constant is related to the values of zeta(2*n-1) of the Riemann zeta function and the Euler Mascheroni constant gamma. If we define Z(n) = (1/n) * (sum(zeta(2*n-2*k-1) * Z(k), k=0..n-2) + gamma * Z(n-1)), with Z(0) = 1, then limit(Z(n), n -> infinity) = 2/exp(1).
Similar formulas appear in A090998 and A112302.
The structure of the n! * Z(n) formulas leads to the multinomial coefficients A036039. (End).

			0.735758882342... = 2*A068985.

Index entries for transcendental numbers

Cf. A036039, A068985, A112302, A090998.

evalf(2/exp(1)) ; # R. J. Mathar, Jul 14 2013

RealDigits[2/E,10,120][[1]] (* Harvey P. Dale, Dec 25 2013 *)

2*exp(-1) \\ Charles R Greathouse IV, Apr 16 2015

Integral of log x from x = 1/e to e. - Charles R Greathouse IV, Apr 16 2015
Equals lim_{k->0} 2*(1 - k)^(1/k). - Ilya Gutkovskiy, Jun 27 2016
Equals Sum_{i>=0} ((-1)^i)(1-i)/i!. - Maciej Kaniewski, Sep 10 2017
Equals Sum_{i>=0} ((-1)^i)(i^2+2)/i!. - Maciej Kaniewski, Sep 12 2017
From Peter Bala, Mar 21 2022: (Start)
2/e = Integral_{x = 1..oo} (2*x/(1+x))^n*(x^2+x+1-n)/x^2*exp(-x) dx;
2/e = - Integral_{x = 0..1} (2*x/(1+x))^n*(x^2+x+1-n)/x^2*exp(-x) dx, both valid for n >= 2. (End)

A163927 Numerators of the higher order exponential integral constants alpha(k,4).

Original entry on oeis.org

1, 49, 1897, 69553, 2515513, 90663937, 3264855049, 117543378001, 4231639039705, 152339702576545, 5484235568128681, 197432536935184369, 7107571838026381177, 255872590744254526273, 9211413307971174616393

Offset: 0

Johannes W. Meijer and Nico Baken, Aug 13 2009, Aug 17 2009

The higher order exponential integrals, see A163931, are defined by E(x,m,n) = x^(n-1)*Integral_{t>=x} E(t,m-1,n)/t^n for m >= 1 and n >= 1, with E(x,m=0,n) = exp(-x).
The series expansions of the higher order exponential integrals are dominated by the alpha(k,n) and the gamma(k,n) constants, see A090998.
The first Maple program uses the alpha(k,n) formula and the second the GF(z,n) to generate the alpha(k,n) coefficients in each column.
Appears to equal the numerator of the multiple harmonic (star) sum Sum_{1 <= k_1 <= ... <= k_n <= 3} 1/(k_1^2*...*k_n^2). If true, then a(n) = numerator( 3/2 - 3/(5*4^n) + 1/(10*9^n) ). - Peter Bala, Jan 31 2019

			a(k=0,n=4) = 1, a(k=1,4) = 49/36, a(k=2,4) = 1897/1296, a(k=3,4) = 69553/46656.

J. W. Meijer and N. H. G. Baken, The Exponential Integral Distribution, Statistics and Probability Letters, Volume 5, No. 3, April 1987, pp. 209-211.

Cf. A163931 (E(x,m,n)), A090998 (gamma(k,n)).
Cf. A163928 y A163929.
a(k,1) = A000007(k)
a(k,2) = A000012(k) = 1^k.
a(k,3) = A002450(k+1)/A000302(k) with A000302(k) = 4^k.
a(k,4) = A163927(k)/A009980(k) with A009980(k) = 36^k.
The GF(z,n) lead to A008955.
The denominators of a(1,n), n >= 2, lead to A007407.

coln := 4; nmax := 15; kmax := nmax: k:=0: for n from 1 to nmax do alpha(k, n) := 1 od: for k from 1 to kmax do for n from 1 to nmax do alpha(k, n) := (1/k)*sum(sum(p^(-2*(k-i)), p=0..n-1)*alpha(i, n), i=0..k-1) od; od: seq(alpha(k, coln), k=0..nmax-1);
# End program 1
coln:=4; nmax1 := 16; for n from 0 to nmax1 do A008955(n, 0):=1 end do: for n from 0 to nmax1 do A008955(n, n) := (n!)^2 end do: for n from 1 to nmax1 do for m from 1 to n-1 do A008955(n, m) := A008955(n-1, m-1)*n^2 + A008955(n-1, m) end do: end do: m:=coln-1: f(m):=0: for n from 0 to m do f(m) := f(m) + (-1)^(n + m)*A008955(m, n)*z^(2*m-2*n) od: GF(z,coln) := m!^2/f(m): GF(z,coln):=series(GF(z,coln), z, nmax1);
# End program 2

alpha(k,n) = (1/k) * Sum_{i=0..k-1} (Sum_{p=0..n-1}(p^(2*i-2*k))*alpha(i, n)) with alpha(0,n) = 1, k >= 0 and n >= 1.
alpha(k,n) = alpha(k,n+1) -alpha(k-1,n+1)/n^2.
GF(z,n) = product((1-(z/k)^2)^(-1), k = 1..n-1) = (Pi*z/sin(Pi*z))/(Beta(n+z,n-z)/Beta(n,n)).

A274760 The multinomial transform of A001818(n) = ((2*n-1)!!)^2.

Original entry on oeis.org

1, 1, 10, 478, 68248, 21809656, 13107532816, 13244650672240, 20818058883902848, 48069880140604832128, 156044927762422185270016, 687740710497308621254625536, 4000181720339888446834235653120, 29991260979682976913756629498334208

Offset: 0

Johannes W. Meijer, Jul 27 2016

nonn

The multinomial transform [MNL] transforms an input sequence b(n) into the output sequence a(n). Given the fact that the structure of the a(n) formulas, see the examples, lead to the multinomial coefficients A036039 the MNL transform seems to be an appropriate name for this transform. The multinomial transform is related to the exponential transform, see A274804 and the third formula. For the inverse multinomial transform [IML] see A274844.
To preserve the identity IML[MNL[b(n)]] = b(n) for n >= 0 for a sequence b(n) with offset 0 the shifted sequence b(n-1) with offset 1 has to be used as input for the MNL, otherwise information about b(0) will be lost in transformation.
In the a(n) formulas, see the examples, the multinomial coefficients A036039 appear.
We observe that a(0) = 1 and that this term provides no information about any value of b(n), this notwithstanding we will start the a(n) sequence with a(0) = 1.
The Maple programs can be used to generate the multinomial transform of a sequence. The first program uses the first formula which was found by Paul D. Hanna, see A158876, and Vladimir Kruchinin, see A215915. The second program uses properties of the e.g.f., see the sequences A158876, A213507, A244430 and A274539 and the third formula. The third program uses information about the inverse multinomial transform, see A274844.
Some MNL transform pairs are, n >= 1: A000045(n) and A244430(n-1); A000045(n+1) and A213527(n-1); A000108(n) and A213507(n-1); A000108(n-1) and A243953(n-1); A000142(n) and A158876(n-1); A000203(n) and A053529(n-1); A000110(n) and A274539(n-1); A000041(n) and A215915(n-1); A000035(n-1) and A177145(n-1); A179184(n) and A038205(n-1); A267936(n) and A000266(n-1); A267871(n) and A000090(n-1); A193356(n) and A088009(n-1).

			Some a(n) formulas, see A036039:
  a(0) = 1
  a(1) = 1*x(1)
  a(2) = 1*x(2) + 1*x(1)^2
  a(3) = 2*x(3) + 3*x(1)*x(2) + 1*x(1)^3
  a(4) = 6*x(4) + 8*x(1)*x(3) + 3*x(2)^2 + 6*x(1)^2*x(2) + 1*x(1)^4
  a(5) = 24*x(5) + 30*x(1)*x(4) + 20*x(2)*x(3) + 20*x(1)^2*x(3) + 15*x(1)*x(2)^2 + 10*x(1)^3*x(2) + 1*x(1)^5

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 1995, pp. 18-23.

M. Bernstein and N. J. A. Sloane, Some Canonical Sequences of Integers, arXiv:math/0205301 [math.CO], 2002; Linear Algebra and its Applications, Vol. 226-228 (1995), pp. 57-72. Erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms.
Eric W. Weisstein MathWorld, Exponential Transform.

Cf. A274844, A274804, A036039, A001818, A001316, A215915, A008279.
Cf. A244430, A213527, A213507, A243953, A158876, A274539, A215915.
Cf. A090998, A008298, A271703, A112302, A135002, A274540.

nmax:= 13: b := proc(n): (doublefactorial(2*n-1))^2 end: a:= proc(n) option remember: if n=0 then 1 else add(((n-1)!/(n-k)!) * b(k) * a(n-k), k=1..n) fi: end: seq(a(n), n = 0..nmax); # End first MNL program.
nmax:=13: b := proc(n): (doublefactorial(2*n-1))^2 end: t1 := exp(add(b(n)*x^n/n, n = 1..nmax+1)): t2 := series(t1, x, nmax+1): a := proc(n): n!*coeff(t2, x, n) end: seq(a(n), n = 0..nmax); # End second MNL program.
nmax:=13: b := proc(n): (doublefactorial(2*n-1))^2 end: f := series(log(1+add(s(n)*x^n/n!, n=1..nmax)), x, nmax+1): d := proc(n): n*coeff(f, x, n) end: a(0) := 1: a(1) := b(1): s(1) := b(1): for n from 2 to nmax do s(n) := solve(d(n)-b(n), s(n)): a(n):=s(n): od: seq(a(n), n=0..nmax); # End third MNL program.

b[n_] := (2*n - 1)!!^2;
a[0] = 1; a[n_] := a[n] = Sum[((n-1)!/(n-k)!)*b[k]*a[n-k], {k, 1, n}];
Table[a[n], {n, 0, 13}] (* Jean-François Alcover, Nov 17 2017 *)

a(n) = Sum_{k=1..n} ((n-1)!/(n-k)!)*b(k)*a(n-k), n >= 1 and a(0) = 1, with b(n) = A001818(n) = ((2*n-1)!!)^2.
a(n) = n!*P(n), with P(n) = (1/n)*(Sum_{k=0..n-1} b(n-k)*P(k)), n >= 1 and P(0) = 1, with b(n) = A001818(n) = ((2*n-1)!!)^2.
E.g.f.: exp(Sum_{n >= 1} b(n)*x^n/n) with b(n) = A001818(n) = ((2*n-1)!!)^2.
denom(a(n)/2^n) = A001316(n); numer(a(n)/2^n) = [1, 1, 5, 239, 8531, 2726207, ...].

A070860 Decimal expansion of Pi^2/12 - gamma^2 /2.

Original entry on oeis.org

6, 5, 5, 8, 7, 8, 0, 7, 1, 5, 2, 0, 2, 5, 3, 8, 8, 1, 0, 7, 7, 0, 1, 9, 5, 1, 5, 1, 4, 5, 3, 9, 0, 4, 8, 1, 2, 7, 9, 7, 6, 6, 3, 8, 0, 4, 7, 8, 5, 8, 4, 3, 4, 7, 2, 9, 2, 3, 6, 2, 4, 4, 5, 6, 8, 3, 8, 7, 0, 8, 3, 8, 3, 5, 3, 7, 2, 2, 1, 0, 2, 0, 8, 6, 1, 8, 2, 8, 1, 5, 9, 9, 4, 0, 2, 1, 3, 6, 4, 0, 0, 0, 4, 8

Offset: 0

Benoit Cloitre, May 24 2003

This is erroneously computed as the linear term in the Laurent expansion of Gamma(x) = 1/x +c(0) + c(1)*x+ O(x^2) on page 135 of the Patterson book. The correct value of c(1) is A090998. - R. J. Mathar, Jul 11 2025

			0.65587807152025388107701951514539048127976638047858434729236244568387...

G. C. Greubel, Table of n, a(n) for n = 0..10000
R. J. Mathar, Erratum to Exercise A4.2 in "An Introduction to the Theory of the Riemann Zeta Function", viXra:2507.0094 (2025)
S. J. Patterson, An introduction to the theory of the Riemann zeta function, Cambridge studies in advanced mathematics no. 14, (1988) p. 135

R:= RealField(100); (Pi(R)^2 - 6*EulerGamma(R)^2)/12; // G. C. Greubel, Sep 05 2018

RealDigits[(Zeta[2] - EulerGamma^2)/2, 10, 100][[1]] (* G. C. Greubel, Sep 05 2018 *)

PARI
```
-(Euler^2-zeta(2))/2
```

(EulerGamma^2 - zeta(2))/2 = -0.65587807152025388....

Equals A072691 - A155969/2.

A163928 Numerators of the higher order exponential integral constants alpha(2,n).

Original entry on oeis.org

0, 1, 21, 1897, 32197, 20881861, 7139587, 17462165587, 283355376967, 69621962857381, 70246946681461, 1036088178214798501, 1042504974775473001, 29931734181763981573561, 4295332813075795410223, 4312254507400142830831

Offset: 1

Johannes W. Meijer & Nico Baken, Aug 13 2009, Aug 17 2009

See A163927 for information about the alpha(k,n) constants.

Apart from a difference of offset, alpha(2,n) appears to be the multiple harmonic (star) sum Sum_{j = 1..n} 1/j^2 Sum_{k = 1..j} 1/k^2, which has the initial values [1, 21/16, 1897/1296, 32197/20736, 20881861/12960000, 7139587/4320000, ...]. - Peter Bala, Jan 31 2019

			alpha(k=2,n=1) = 0, alpha(k=2,2) = 1, alpha(k=2,3) = 21/16, alpha(k=2,4) = 1897/1296.

Cf. A163929 (denominators).
Cf. A163927 (alpha(k,n)) and A090998 (gamma(k,n)).
Cf. A007406, A007407.

nmax:=17; rowk:=2; kmax:=nmax: k:=0: for n from 1 to nmax do alpha(k,n):=1 od: for k from 1 to kmax do for n from 1 to nmax do alpha(k,n) := (1/k)*sum(sum(p^(-2*(k-i)),p=0..n-1)*alpha(i, n),i=0..k-1) od; od: seq(alpha(rowk, n),n=1..nmax);

alpha(k,n) = (1/k)*Sum_{i=0..k-1} (Sum_{p=0..n-1} p^(-2*(k-i))*alpha(i, n) with alpha(0,n) = 1, with k = 2 and n >= 1. alpha(1,n) = A007406(n-1)/A007407(n-1) for n >= 2.

A385960 Decimal expansion of the absolute value of the coefficient [x^2] Gamma(x).

Original entry on oeis.org

9, 0, 7, 4, 7, 9, 0, 7, 6, 0, 8, 0, 8, 8, 6, 2, 8, 9, 0, 1, 6, 5, 6, 0, 1, 6, 7, 3, 5, 6, 2, 7, 5, 1, 1, 4, 9, 2, 8, 6, 1, 1, 4, 4, 9, 0, 7, 2, 5, 6, 3, 7, 6, 0, 9, 4, 1, 3, 3, 1, 1, 5, 4, 0, 5, 0, 4, 6, 5, 1, 8, 2, 3, 7, 2, 2, 3, 0, 6, 9, 3, 9, 8, 3, 8, 7, 5, 2, 7, 4, 1, 1, 3, 6, 2, 9, 7, 7, 2, 1, 6, 8, 2, 1

Offset: 0

R. J. Mathar, Jul 13 2025

The Laurent series Gamma(x) = 1/x + Sum_{i>=0} a_i x^i starts with a_0 = -gamma = -A001620, a_1 = A090998 , and a_2 = -0.90747907.. , absolute value here. Recurrence (i+1)*a_i = -gamma *a_{i-1} + Sum_{k=2..i+1} (-1)^k*zeta(k)a_{i-k} .

			0.9074790760808862890165601673...

I. S. Gradsteyn, I. M. Ryzhik, Tables of Series and Products, Academic Press (2014) 8.321.1
R. J. Mathar, Erratum to Exercise A4.2 in "An Introduction to the Theory of the Riemann Zeta Function", viXra:2507.0094 (2025)

Cf. A090998 [x^1], A385961 [x^3], A385962 [x^4].

(gamma^3+3*gamma*Zeta(2)+2*Zeta(3))/6 ; evalf(%) ;

polcoef(gamma(x), 2) \\ Michel Marcus, Jul 13 2025

Equals (gamma^3 +3*gamma*zeta(2) +2*zeta(3))/6 , gamma = A001620, zeta(2) = A013661, zeta(3)=A002117.

A385961 Decimal expansion of the value of the coefficient [x^3] Gamma(x).

Original entry on oeis.org

9, 8, 1, 7, 2, 8, 0, 8, 6, 8, 3, 4, 4, 0, 0, 1, 8, 7, 3, 3, 6, 3, 8, 0, 2, 9, 4, 0, 2, 1, 8, 5, 0, 8, 5, 0, 3, 6, 0, 5, 7, 3, 6, 7, 9, 7, 2, 3, 4, 6, 5, 4, 1, 5, 4, 0, 4, 9, 5, 7, 4, 5, 5, 5, 9, 3, 8, 5, 6, 8, 3, 9, 2, 4, 8, 6, 9, 3, 4, 5, 0, 9, 4, 1, 0, 5, 9, 7, 7, 0, 5, 1, 8, 7, 5, 7, 0, 6, 5, 9, 5, 5, 8, 8, 5, 0, 6, 7, 0, 4, 3, 6, 8, 2

Offset: 0

R. J. Mathar, Jul 13 2025

The Laurent series Gamma(x) = 1/x + Sum_{i>=0} a_i x^i starts with a_0 = -gamma = -A001620, a_1 = A090998 . a_3 = 0.9817280868.. is here.

			0.981728086834400187336380294021850...

I. S. Gradsteyn, I. M. Ryzhik, Tables of Series and Products, Academic Press (2014) 8.321.1 gives recurrence.
R. J. Mathar, Erratum to Exercise A4.2 in "An Introduction to the Theory of the Riemann Zeta Function", viXra:2507.0094 (2025)

Cf. A090998 [x^1], A385960 [x^2], A385962 [x^4].

(gamma^4+6*gamma^2*Zeta(2)+8*gamma*Zeta(3)+3*Zeta(2)^2+6*Zeta(4))/24 ; evalf(%) ;

Equals (gamma^4 +6*gamma^2*zeta(2) +8*gamma*zeta(3) +3*zeta(2)^2 +6*zeta(4))/24 , gamma = A001620, zeta(2) = A013661, zeta(3)=A002117, zeta(4) = A013662.

cp's OEIS Frontend

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A274760 The multinomial transform of A001818(n) = ((2*n-1)!!)^2.

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A070860 Decimal expansion of Pi^2/12 - gamma^2 /2.

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