A133942 -id:A133942 - cp's OEIS frontend

A177700 The n-th derivative of log(1+x)*tanh(x) evaluated at x = 0.

0, 0, 2, -3, 0, -10, 160, -756, 2688, -27504, 341248, -3113440, 29004800, -365574144, 5120567296, -69912541440, 1009388355584, -16301637449728, 281310403362816, -5030932957138944, 94747161802047488, -1897026741117419520

Offset: 0

Michel Lagneau, May 11 2010

sign

			The second derivative is -(tanh(x)/(x+1)^2) + 2*((1 - tanh(x)^2)/(x+1)) - 2*log(x+1)tanh(x)(1 - tanh(x)^2). At x = 0 this sets a(2) = 0 + 2 - 0 = 2.

L. Comtet and M. Fiolet, Sur les dérivées successives d'une fonction implicite. C. R. Acad. Sci. Paris Ser. A 278 (1974), 249-251. MR0348055

Cf. A177699, A133942 (derivatives of log(1+x)), A155585 (derivatives of tanh(x)).

n0:= 35: T:=array(1..n0): f:=x-> ln(1+x)*tanh(x):
for n from 1 to n0 do: T[n]:=D(f)(0):f:=D(f):od: print(T):

a(0) inserted and keyword:sign added by R. J. Mathar, May 14 2010

A305786 Inverse Euler transform of (-1)^n * n!.

Original entry on oeis.org

-1, 2, -4, 17, -92, 576, -4156, 34159, -314368, 3199936, -35703996, 433421495, -5687955724, 80256879068, -1211781887796, 19496946534720, -333041104402860, 6019770247224496, -114794574818830716, 2303332661416242633, -48509766592884311132, 1069983257387168051076

Offset: 1

Seiichi Manyama, Jun 10 2018

sign

			(1-x) * (1-x^2)^(-2) * (1-x^3)^4 * (1-x^4)^(-17) * ... = 1 - x + 2*x^2 - 6*x^3 + 24*x^4 - ... .

Seiichi Manyama, Table of n, a(n) for n = 1..449
N. J. A. Sloane, Transforms

Cf. A112354, A133942.

Product_{k>=1} 1/(1-x^k)^{a(k)} = Sum_{n>=0} (-1)^n * n! * x^n.

a(n) ~ (-1)^n * n!. - Vaclav Kotesovec, Oct 09 2019

A321398 a(n) = (-1)^(n+1)n! [x^n](log(x + 1)/2 + log(3*x + 1)/6).

Original entry on oeis.org

0, 1, 2, 10, 84, 984, 14640, 262800, 5513760, 132289920, 3571464960, 107140320000, 3535590643200, 127280784153600, 4963944354969600, 208485575730432000, 9381849600195072000, 450328759886573568000, 22966766398527823872000, 1240205379118128783360000

Offset: 0

Peter Luschny, Nov 10 2018

nonn

Robert Israel, Table of n, a(n) for n = 0..381

Cf. A133942 (n=1), A000165 (n=2), this sequence (n=3), A320962 (limit).

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( -Log((1-x)^3*(1-3*x))/6 )); [0] cat [Factorial(n-0)*b[n]: n in [1..(m-1)]]; // G. C. Greubel, Nov 11 2018

ser := series(ln(x+1)/2 + ln(1+3*x)/6, x, 21):
seq((-1)^(n+1)*n!*coeff(ser, x, n), n=0..19);

CoefficientList[Series[Log[x+1]/2 + Log[1+3*x]/6, {x, 0, 50}], x]* Table[(-1)^(n+1)*n!, {n, 0, 50}] (* Stefano Spezia, Nov 10 2018 *)

seq(n)={Vec(serlaplace(-log(1 - x + O(x^n))/2 - log(1 - 3*x + O(x^n))/6), -n)} \\ Andrew Howroyd, Nov 10 2018

E.g.f.: -log(1 - x)/2 - log(1 - 3*x)/6. - Andrew Howroyd, Nov 10 2018

3*n*(n+1)*a(n)-4*(n+1)*a(n)+a(n+2)=0. - Robert Israel, Nov 10 2018

A331725 E.g.f.: exp(x/(1 - x)) / (1 + x).

Original entry on oeis.org

1, 0, 3, 4, 57, 216, 2755, 18348, 247569, 2368432, 35256771, 436248660, 7235178313, 108919083144, 2010150360387, 35421547781116, 723689454172065, 14543895730321248, 326843345169621379, 7354350135365751972, 180610925178770615001, 4488323611011676811320

Offset: 0

Ilya Gutkovskiy, Jan 25 2020

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..444

Cf. A000262, A002720, A009940, A133942, A182386, A206703.

A331725 := proc(n)
    add((-1)^k*binomial(n,k)*k!*A000262(n-k),k=0..n) ;
end proc:
seq(A331725(n),n=0..42) ; # R. J. Mathar, Aug 20 2021

nmax = 21; CoefficientList[Series[Exp[x/(1 - x)]/(1 + x), {x, 0, nmax}], x] Range[0, nmax]!
A000262[n_] := If[n == 0, 1, n! Sum[Binomial[n - 1, k]/(k + 1)!, {k, 0, n - 1}]]; a[n_] := Sum[(-1)^k Binomial[n, k] k! A000262[n - k], {k, 0, n}]; Table[a[n], {n, 0, 21}]
a[n_] := (-1)^n n! (1 - Sum[(-1)^j*LaguerreL[j, 1, -1]/(j+1), {j,0,n-1}]);
Table[a[n], {n, 0, 21}] (* Peter Luschny, Feb 20 2022 *)

seq(n)={Vec(serlaplace(exp(x/(1 - x) + O(x*x^n)) / (1 + x)))} \\ Andrew Howroyd, Jan 25 2020

def gen_a():
    F, L, S, N = 1, 1, 1, 1
    while True:
        yield F * S
        L = gen_laguerre(N - 1, 1, -1) / N
        S += L if F < 0 else -L
        F *= -N; N += 1
a = gen_a(); print([next(a) for  in range(21)]) # _Peter Luschny, Feb 20 2022

a(n) = Sum_{k=0..n} (-1)^k * binomial(n,k) * k! * A000262(n-k).
a(n) ~ n^(n - 1/4) / (2^(3/2) * exp(1/2 - 2*sqrt(n) + n)). - Vaclav Kotesovec, Jan 26 2020
D-finite with recurrence a(n) +(-n+1)*a(n-1) -(n-1)*(n+1)*a(n-2) +(n-1)*(n-2)^2*a(n-3)=0. - R. J. Mathar, Aug 20 2021
a(n) = Sum_{k=0..n} (-1)^k * A206703(n,k). - Alois P. Heinz, Feb 19 2022
a(n) = (-1)^n*n!*(1 - Sum_{j=0..n-1}((-1)^j*LaguerreL(j, 1, -1)/(j + 1))). - Peter Luschny, Feb 20 2022

A265313 Square array read by ascending antidiagonals, complementary Bell numbers iterated by the Bell transform.

Original entry on oeis.org

1, 1, 1, 1, -1, 1, 1, -1, 0, 1, 1, -1, 2, 1, 1, 1, -1, 2, -4, 1, 1, 1, -1, 2, -6, 9, -2, 1, 1, -1, 2, -6, 22, -22, -9, 1, 1, -1, 2, -6, 24, -95, 54, -9, 1, 1, -1, 2, -6, 24, -118, 472, -139, 50, 1, 1, -1, 2, -6, 24, -120, 683, -2638, 372, 267, 1, 1, -1, 2, -6, 24

Offset: 0

Peter Luschny, Dec 06 2015

			[ 1,  1, 1,  1,  1,   1,    1,     1,     1, ...] A000012
[ 1, -1, 0,  1,  1,  -2,   -9,    -9,    50, ...] A000587
[ 1, -1, 2, -4,  9, -22,   54,  -139,   372, ...] A265023
[ 1, -1, 2, -6, 22, -95,  472, -2638, 16343, ...]
[ 1, -1, 2, -6, 24, -118, 683, -4533, 33862, ...]
[ 1, -1, 2, -6, 24, -120, 718, -4989, 39405, ...]
[...                                         ...]
[ 1, -1, 2, -6, 24, -120, 720, -5040, 40320, ...] A133942

Peter Luschny, The Bell transform

Cf. A000012, A000587, A133942, A264428, A265023, A265312.

# uses[bell_transform from A264428]
def complementary_bell_number_matrix(ord, len):
    b = [1]*len; L = [b]
    for k in (1..ord-1):
        b = [sum((-1)^n*c for (n, c) in enumerate(bell_transform(n, b))) for n in range(len)]
        L.append(b)
    return matrix(ZZ, L)
print(complementary_bell_number_matrix(6,9))

A302190 Hurwitz logarithm of natural numbers 1,2,3,4,5,...

Original entry on oeis.org

0, 2, -1, 2, -6, 24, -120, 720, -5040, 40320, -362880, 3628800, -39916800, 479001600, -6227020800, 87178291200, -1307674368000, 20922789888000, -355687428096000, 6402373705728000, -121645100408832000, 2432902008176640000, -51090942171709440000

Offset: 0

N. J. A. Sloane and William F. Keigher, Apr 12 2018

sign

In the ring of Hurwitz sequences all members have offset 0.

Xing Gao and William F. Keigher, Interlacing of Hurwitz series, Communications in Algebra, 45:5 (2017), 2163-2185, DOI: 10.1080/00927872.2016.1226885. See Ex. 2.16.
N. J. A. Sloane, Maple programs for operations on Hurwitz sequences

Cf. A133942.

# first load Maple commands for Hurwitz operations from link
s:=[seq(n,n=1..64)];
Hlog(s);

A = PowerSeriesRing(QQ, 'x')
f = A(list(range(1,30))).ogf_to_egf().log()
print(list(f.egf_to_ogf()))
# F. Chapoton, Apr 11 2020

E.g.f. is log of Sum_{n >= 0} (n+1)*x^n/n!.

A355007 Triangle read by rows. T(n, k) = n^k * |Stirling1(n, k)|.

Original entry on oeis.org

1, 0, 1, 0, 2, 4, 0, 6, 27, 27, 0, 24, 176, 384, 256, 0, 120, 1250, 4375, 6250, 3125, 0, 720, 9864, 48600, 110160, 116640, 46656, 0, 5040, 86436, 557032, 1764735, 2941225, 2470629, 823543, 0, 40320, 836352, 6723584, 27725824, 64225280, 84410368, 58720256, 16777216

Offset: 0

Peter Luschny, Jun 17 2022

			Table T(n, k) begins:
[0] 1;
[1] 0,    1;
[2] 0,    2,     4;
[3] 0,    6,    27,     27;
[4] 0,   24,   176,    384,     256;
[5] 0,  120,  1250,   4375,    6250,    3125;
[6] 0,  720,  9864,  48600,  110160,  116640,   46656;
[7] 0, 5040, 86436, 557032, 1764735, 2941225, 2470629, 823543;

A000142 (column 1), A000407 (row sums), A000312 (main diagonal), A355006.

Cf. A133942.

seq(seq(n^k*abs(Stirling1(n, k)), k = 0..n), n = 0..9);

T[n_, k_] := If[n == k == 0, 1, n^k * Abs[StirlingS1[n, k]]]; Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Amiram Eldar, Jun 17 2022 *)

Sum_{k=0..n} (-1)^k * T(n,k) = A133942(n). - Alois P. Heinz, Mar 30 2023

Conjecture: T(n,k) = A056856(n,k)*n. - R. J. Mathar, Mar 31 2023

A138160 A triangular sequence of coefficients of the expansion of a Green's function for the radial Morse potential with x being the kinetic energy and t being the radius: Hamiltonian; H=K+V=x+Exp[-2t]-2Exp[t];GExp[tx)=Exp[xt]/(t-H); p(t,x)=Exp[tx]/(t-x-Exp[-2t] + 2Exp[-t]).

Original entry on oeis.org

1, -1, 1, -1, 4, -4, 3, -2, 1, -24, 36, -27, 13, -6, 3, -1, 182, -354, 330, -198, 85, -28, 10, -4, 1, -1730, 4090, -4480, 3120, -1595, 631, -195, 50, -15, 5, -1, 19802, -55270, 70430, -55730, 31630, -14018, 5101, -1536, 375, -80, 21, -6, 1, -264334, 850990, -1246504, 1121960, -711480, 345268, -135639, 44997, -12922, 3171, -644, 119, -28, 7, -1

Offset: 1

Roger L. Bagula, May 04 2008

The row sums are:A133942; (-1)^n*n!;
{1, -1, 2, -6, 24, -120, 720, -5040, 40320, -362880, 3628800};
Plotting the relative Kinetic curves gives even peaks and odd Serpentine curves:
s = Table[Plot[g[[n]]/(1 - x)^(n + 1), {x, 0, 10}], {n, 1, Length[g]}].
It appears this method solves the spectrum of relative excited states for a Morse potential Hamiltonian system.

			Triangle begins:
  {1},
  {-1, 1, -1},
  {4, -4, 3, -2, 1},
  {-24, 36, -27, 13, -6, 3, -1},
  {182, -354, 330, -198, 85, -28, 10, -4, 1},
  {-1730, 4090, -4480, 3120, -1595, 631, -195, 50, -15, 5, -1},
  {19802, -55270, 70430, -55730, 31630, -14018, 5101, -1536, 375, -80, 21, -6, 1},
  {-264334, 850990, -1246504, 1121960, -711480, 345268, -135639, 44997, -12922, 3171, -644, 119, -28, 7, -1},
  ...

A. Messiah, Quantum mechanics, vol. 2, page 712, 795-800, North Holland, 1969.

p[t_, x_] = FullSimplify[Exp[t*x]/(t - x - Exp[ -2*t] + 2*Exp[ -t])] g = Table[ FullSimplify[ExpandAll[n!*(1 - x)^(n + 1)*SeriesCoefficient[ Series[p[t, x], {t, 0, 30}], n]]], {n, 0, 10}]; a = Table[CoefficientList[FullSimplify[ExpandAll[n!*(1 - x)^(n + 1)*SeriesCoefficient[Series[p[t, x], {t, 0, 30}], n]]], x], {n, 0, 10}]; Flatten[a]

p(t,x)=Exp[t*x]/(t - x - Exp[ -2*t] + 2*Exp[ -t])=sum(P(x,n)*t^n/n!,m{n,0,Infinity}); out_n,m=Coefficients(n!*(1-x)^(n+1)*P(x,n)).

A165233 Signed denominators of terms in series expansion of cos(x)+sin(x).

Original entry on oeis.org

1, 1, -2, -6, 24, 120, -720, -5040, 40320, 362880, -3628800, -39916800, 479001600, 6227020800, -87178291200, -1307674368000, 20922789888000, 355687428096000, -6402373705728000, -121645100408832000, 2432902008176640000

Offset: 0

Jaume Oliver Lafont, Sep 09 2009

Sum(n>=0,1/a(n))=cos(1)+sin(1).
Sum(n>=0,(Pi/4)^n/a(n))=sqrt(2).
Numerators are in A000012. - Alois P. Heinz, Jan 20 2016

Cf. A000012, A000142, A133942, A057077, A143623, A002193.

Sign@ # Denominator@ # & /@ CoefficientList[Series[Cos@ x + Sin@ x, {x, 0, 20}], x] (* Michael De Vlieger, Oct 08 2016 *)

PARI
```
a(n)=(-1)^(n\2)*n!
```

a(n)=(-1)^floor(n/2)*n! = A057077(n)*A000142(n).
a(n)=(sin(n*Pi/2)+cos(n*Pi/2))*n!.
a(n)=sqrt(2)*sin((2n+1)*Pi/4)*n!.
a(n)=sqrt(2)*cos((2n-1)*Pi/4)*n!.
G.f. Q(0) where Q(k)= 1 + x*(4*k+1)/(1 + 2*x*(2*k+1)/(1 - 2*x*(2*k+1) - x*(4*k+3)/(1 + x*(4*k+3) - 4*x*(k+1)/(4*x*(k+1) - 1/Q(k+1))))); (continued fraction, 3rd kind, 5-step). - Sergei N. Gladkovskii, Aug 15 2012
E.g.f.: (1 + x)/(1 + x^2). - Ilya Gutkovskiy, Oct 08 2016

A358624 Triangle read by rows. The coefficients of the Hahn polynomials in ascending order of powers. T(n, k) = n! * [x^k] hypergeom([-x, -n, n + 1], [1, 1], 1).

Original entry on oeis.org

1, 1, 2, 2, 6, 6, 6, 22, 30, 20, 24, 100, 170, 140, 70, 120, 548, 1050, 1120, 630, 252, 720, 3528, 7476, 8820, 6720, 2772, 924, 5040, 26136, 59388, 78708, 64680, 37884, 12012, 3432, 40320, 219168, 529896, 748440, 704550, 432432, 204204, 51480, 12870

Offset: 0

Peter Luschny, Nov 26 2022

			[0]     1;
[1]     1,      2;
[2]     2,      6,      6;
[3]     6,     22,     30,     20;
[4]    24,    100,    170,    140,     70;
[5]   120,    548,   1050,   1120,    630,    252;
[6]   720,   3528,   7476,   8820,   6720,   2772,    924;
[7]  5040,  26136,  59388,  78708,  64680,  37884,  12012,  3432;
[8] 40320, 219168, 529896, 748440, 704550, 432432, 204204, 51480, 12870;

A. F. Nikiforov, S. K. Suslov, and V. B. Uvarov, Classical Orthogonal Polynomials of a Discrete Variable, Springer-Verlag Berlin Heidelberg, 1991.

Cf. A000142, A000984, A001564 (row sums), A133942 (alternating row sums).

H := (n, x) -> n!*hypergeom([-x, -n, n + 1], [1, 1], 1):
for n from 0 to 8 do seq(coeff(simplify(H(n, x)), x, k), k = 0..n) od;

The general formula for the Hahn polynomials is H(n, x, N, a, b) = (-1)^n*(Pochhammer(N-n, n)*Pochhammer(b+1, n) / n!)*hypergeom([-n, -x, a + b + n + 1], [b + 1, 1 - N], 1). We consider here the case N = a = b = 0.

cp's OEIS Frontend

A177700 The n-th derivative of log(1+x)*tanh(x) evaluated at x = 0.

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A305786 Inverse Euler transform of (-1)^n * n!.

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A321398 a(n) = (-1)^(n+1)*n!* [x^n](log(x + 1)/2 + log(3*x + 1)/6).

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A331725 E.g.f.: exp(x/(1 - x)) / (1 + x).

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A265313 Square array read by ascending antidiagonals, complementary Bell numbers iterated by the Bell transform.

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Sage

A302190 Hurwitz logarithm of natural numbers 1,2,3,4,5,...

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A355007 Triangle read by rows. T(n, k) = n^k * |Stirling1(n, k)|.

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A138160 A triangular sequence of coefficients of the expansion of a Green's function for the radial Morse potential with x being the kinetic energy and t being the radius: Hamiltonian; H=K+V=x+Exp[-2*t]-2*Exp[t];G*Exp[t*x)=Exp[x*t]/(t-H); p(t,x)=Exp[t*x]/(t-x-Exp[-2*t] + 2*Exp[-t]).

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A321398 a(n) = (-1)^(n+1)n! [x^n](log(x + 1)/2 + log(3*x + 1)/6).

A138160 A triangular sequence of coefficients of the expansion of a Green's function for the radial Morse potential with x being the kinetic energy and t being the radius: Hamiltonian; H=K+V=x+Exp[-2t]-2Exp[t];GExp[tx)=Exp[xt]/(t-H); p(t,x)=Exp[tx]/(t-x-Exp[-2t] + 2Exp[-t]).