A003319 -id:A003319 - cp's OEIS frontend

A090365 Shifts 1 place left under the INVERT transform of the BINOMIAL transform of this sequence.

1, 1, 3, 11, 47, 225, 1177, 6625, 39723, 251939, 1681535, 11764185, 86002177, 655305697, 5193232611, 42726002123, 364338045647, 3215471252769, 29331858429241, 276224445794785, 2682395337435723, 26832698102762435, 276221586866499839, 2923468922184615897

Offset: 0

Paul D. Hanna, Nov 26 2003

nonn

The Hankel transform of this sequence is A000178(n+1); example: det([1,1,3; 1,3,11; 3,11,47]) = 12. - Philippe Deléham, Mar 02 2005
a(n) appears to be the number of indecomposable permutations (A003319) of [n+1] that avoid both of the dashed patterns 32-41 and 41-32. - David Callan, Aug 27 2014
This is true: A nonempty permutation avoids 32-41 and 41-32 if and only if all its components do so. So if A(x) denotes the g.f. for indecomposable {32-41,41-32}-avoiders, then F(x):=1/(1-A(x)) is the g.f. for all {32-41,41-32}-avoiders. From A074664, F(x)=1/x(1-1/B(x)) where B(x) is the o.g.f. for the Bell numbers. Solve for A(x). - David Callan, Jul 21 2017
The Hankel transform of this sequence without the a(0)=1 term is also A000178(n+1). - Michael Somos, Oct 02 2024

Alois P. Heinz, Table of n, a(n) for n = 0..574

Cf. A090366, A090367.
Cf. A074664.
Similar recurrences: A006014, A124758, A217924, A243499, A284005, A290615, A329369, A341392, A347204, A347205.

bintr:= proc(p) proc(n) add(p(k) *binomial(n,k), k=0..n) end end:
invtr:= proc(p) local b;
           b:= proc(n) option remember; local i;
                `if`(n<1, 1, add(b(n-i) *p(i-1), i=1..n+1))
               end;
        end:
b:= invtr(bintr(a)):
a:= n-> `if`(n<0, 0, b(n-1)):
seq(a(n), n=0..30);  # Alois P. Heinz, Jun 28 2012

a[n_] := Module[{A, B}, A = 1+x; For[k=1, k <= n, k++, B = (A /. x -> x/(1 - x))/(1-x) + O[x]^n // Normal; A = 1 + x*A*B]; SeriesCoefficient[A, {x, 0, n}]]; Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Oct 23 2016, adapted from PARI *)

{a(n)=local(A); if(n<0,0,A=1+x+x*O(x^n); for(k=1,n,B=subst(A,x, x/(1-x))/(1-x)+x*O(x^n); A=1+x*A*B);polcoeff(A,n,x))}

G.f.: A(x) satisfies A(x) = 1/(1 - A(x/(1-x))*x/(1-x) ).
a(n) = Sum_{k = 0..n} A085838(n, k). - Philippe Deléham, Jun 04 2004
G.f.: 1/x-1-1/(B(x)-1) where B(x) = g.f. for A000110 the Bell numbers. - Vladeta Jovovic, Aug 08 2004
a(n) = Sum_{k=0..n} A094456(n,k). - Philippe Deléham, Nov 07 2007
G.f.: 1/(1-x/(1-2x/(1-x/(1-3x/(1-x/(1-4x/(1-x/(1-5x/(1-... (continued fraction). - Paul Barry, Feb 25 2010
From Sergei N. Gladkovskii, Jan 06 2012  - May 12 2013: (Start)
Continued fractions:
G.f.: 1 - x/(G(0)+x); G(k) = x - 1 + x*k + x*(x-1+x*k)/G(k+1).
G.f.: 1/x - 1/2 + (x^2-4)/(4*U(0)-2*x^2+8) where U(k) = k*(2*k+3)*x^2 + x - 2 - (2-x+2*k*x)*(2+3*x+2*k*x)*(k+1)*x^2/U(k+1).
G.f.: 1/x+1/(U(0)-1) where U(k) = -x*k + 1 - x - x^2*(k+1)/U(k+1).
G.f.: (1 - U(0))/x - 1 where U(k) = 1 - x*(k+2) - x^2*(k+1)/U(k+1).
G.f.: (1 - U(0))/x where U(k) = 1 - x*(k+1)/(1-x/U(k+1)).
G.f.: 1/x + 1/( G(0)-1) where G(k) = 1 - x/(1 - x*(2*k+1)/(1 - x/(1 - x*(2*k+2)/ G(k+1) ))).
G.f.:1/x + 1/( G(0) - 1 ) where G(k) = 1 - x/(1 - x*(k+1)/G(k+1) ).
G.f.: (1 - Q(0))/x where Q(k) = 1 + x/(x*k - 1 )/Q(k+1).
G.f.: 1/x - 1/x/Q(0), where Q(k) = 1 + x/(1 - x + x*(k+1)/(x - 1/Q(k+1))).
(End)
Conjecture: a(n) = b(2^(n-1) - 1) for n > 0 with a(0) = 1 where b(n) = b((n - 2^f(n))/2) + b(floor((2n - 2^f(n))/2)) + b(A025480(n-1)) for n > 0 with b(0) = 1 and where f(n) = A007814(n). - Mikhail Kurkov, Jan 11 2022

A104981 Column 1 of triangle A104980; also equals column 0 of triangle A104986, which equals the matrix logarithm of A104980.

Original entry on oeis.org

0, 1, 2, 7, 33, 191, 1297, 10063, 87669, 847015, 8989301, 103996703, 1303132269, 17589153719, 254509227541, 3931158238735, 64573130459613, 1124144767682215, 20677664894412965, 400760695386194687, 8163539437728923181

Offset: 0

Paul D. Hanna, Apr 10 2005

nonn

G. C. Greubel, Table of n, a(n) for n = 0..440

Cf. A104980, A104982, A104986, A128175.

T[n_, k_]:= T[n, k]= If[nJean-François Alcover, Aug 09 2018 *)

{a(n) = if(n<0, 0, (matrix(n+2, n+2, m, j, if(m==j, 1, if(m==j+1, -m+1, -polcoeff((1-1/sum(i=0, m, i!*x^i))/x +O(x^m), m-j-1))))^-1)[n+1,2])}

@CachedFunction
def T(n,k):
    if (k<0 or k>n): return 0
    elif (k==n): return 1
    elif (k==n-1): return n
    else: return k*T(n, k+1) + sum( T(j, 0)*T(n, j+k+1) for j in (0..n-k-1) )
[T(n,1) for n in (0..30)] # G. C. Greubel, Jun 07 2021

From Gary W. Adamson, Jul 14 2011: (Start)
Let M = triangle A128175 as an infinite square production matrix (deleting the first "1"):
  1, 1, 0, 0, 0, ...
  2, 2, 1, 0, 0, ...
  4, 4, 3, 1, 0, ...
  8, 8, 7, 4, 1, ...
  ...
a(n) = sum of top row terms of M^(n-1). Example: top row of M^4 = (71, 71, 38, 10, 1), sum = 191 = a(5). (End)
a(0) = 1, a(n) = n * a(n-1) + Sum_{j=1..n} A003319(j) * a(n - j), with offset 0 for the term 1. - F. Chapoton, Feb 26 2018

A111534 Main diagonal of table A111528.

Original entry on oeis.org

1, 1, 4, 33, 416, 7045, 149472, 3804353, 112784896, 3812791581, 144643185600, 6081135558817, 280510445260800, 14080668974435141, 763890295406672896, 44529851124925034625, 2775373003913373810688, 184147301185264051623181

Offset: 0

Paul D. Hanna, Aug 06 2005

nonn

For n>0, a(n) is divisible by n: a(n)/n = A111535(n).

Cf: A111528 (table), A003319 (row 1), A111529 (row 2), A111530 (row 3), A111531 (row 4), A111532 (row 5), A111533 (row 6).

T[n_, k_] := T[n, k] = Which[n<0 || k<0, 0, k==0 || k==1, 1, n==0, k!, True, (T[n-1, k+1]-T[n-1, k])/n - Sum[T[n, j] T[n-1, k-j], {j, 1, k-1}]];
a[n_] := T[n, n];
Table[a[n], {n, 0, 17}] (* Jean-François Alcover, Aug 09 2018 *)

{a(n)=if(n<0,0,if(n==0,1, polcoeff(log(sum(m=0,n,(n-1+m)!/(n-1)!*x^m)),n)))}

a(n) = [x^n] Log( Sum_{m=0..n} (n-1+m)!/(n-1)!*x^m ).

A112354 Inverse Euler transform of n!. Also the number of sequences of permutations with no global descents which are Lyndon (smallest in lexicographic order of all cyclic shifts of the sequences) where the size of the sequence = sum of sizes of the permutations.

Original entry on oeis.org

1, 1, 4, 17, 92, 572, 4156, 34159, 314368, 3199844, 35703996, 433421495, 5687955724, 80256874912, 1211781887796, 19496946534720, 333041104402860, 6019770246910128, 114794574818830716, 2303332661416242633, 48509766592884311132, 1069983257387132347080

Offset: 1

Mike Zabrocki, Sep 05 2005

nonn

			a(3) = 4 because (123), (213), (132) and (1,21) are all Lyndon.
a(4) = 17 because there are 13 permutations with no global descents of size 4 and (1,123), (1,213), (1,132) are all Lyndon.
a(5) = 92 = 71 permutations with no global descents+13 sequences of the form (1,pi) where pi in S_4 with no global descents+(1,1,1,21),(1,21,21),(1,1,123),(1,1,213),(1,1,132),(21,123),(21,213),(21,132).

Seiichi Manyama, Table of n, a(n) for n = 1..449 (terms 1..200 from Alois P. Heinz)
M. Aguiar and A. Lauve, Antipode and Convolution Powers of the Identity in Graded Connected Hopf Algebras, FPSAC 2013 Paris, France DMTCS Proc. AS, 2013, 1083-1094.
M. Aguiar, A. Lauve, The characteristic polynomial of the Adams operators on graded connected Hopf algebras, arXiv:1403.7584 [math.RA], 2014.

Cf. A003319, A000142, A305786.

read transforms; EULERi([seq(n!,n=1..30)]);
# The function EulerInvTransform is defined in A358451.
a := EulerInvTransform(factorial):
seq(a(n), n = 1..22); # Peter Luschny, Nov 21 2022

ff = Range[n = 22]!; s = {}; For[i = 1, i <= n, i++, AppendTo[s, i*ff[[i]] - Sum[s[[d]]*ff[[i-d]], {d, i-1}]]]; Table[Sum[If[Divisible[i, d], MoebiusMu[i/d], 0]*s[[d]], {d, 1, i}]/i, {i, n}] (* Jean-François Alcover, Apr 15 2016 *)

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

a(n) ~ n! * (1 - 1/n - 1/n^2 - 4/n^3 - 23/n^4 - 171/n^5 - 1542/n^6 - 16241/n^7 - 194973/n^8 - 2622610/n^9 - 39027573/n^10 - ...), for coefficients see A113869. - Vaclav Kotesovec, Sep 04 2014, extended Nov 27 2020

A158882 G.f. A(x) satisfies: [x^n] A(x)^n = [x^n] A(x)^(n-1) for n>1 with A(0)=A'(0)=1.

Original entry on oeis.org

1, 1, -1, 3, -13, 71, -461, 3447, -29093, 273343, -2829325, 31998903, -392743957, 5201061455, -73943424413, 1123596277863, -18176728317413, 311951144828863, -5661698774848621, 108355864447215063, -2181096921557783605

Offset: 0

Paul D. Hanna, Apr 30 2009

sign

After initial term, equals signed A003319 (indecomposable permutations).

			G.f.: A(x) = 1 + x - x^2 + 3*x^3 - 13*x^4 + 71*x^5 - 461*x^6 +-...
1/A(x) = 1 - x + 2*x^2 - 6*x^3 + 24*x^4 +...+ (-1)^n*n!*x^n +...
...
Coefficients of powers of g.f. A(x) begin:
A^1: 1,1,(-1),3,-13,71,-461,3447,-29093,273343,-2829325,...;
A^2: 1,2,(-1),(4),-19,110,-745,5752,-49775,476994,-5016069,...;
A^3: 1,3, 0, (4),(-21),129,-910,7242,-64155,626319,-6685548,...;
A^4: 1,4, 2, 4, (-21),(136),-996,8152,-73811,733244,-7938186,...;
A^5: 1,5, 5, 5, -20, (136),(-1030),8650,-79925,807055,-8854741,...;
A^6: 1,6, 9, 8, -18, 132, (-1030),(8856),-83385,855010,-9500385,...;
A^7: 1,7,14,14, -14, 126, -1008, (8856),(-84861),882805,-9927890,...;
A^8: 1,8,20,24, -6, 120, -972, 8712, (-84861),(894928),-10180120,...;
A^9: 1,9,27,39,9,117,-927,8469,-83772,(894928),(-10291986),...;
A^10:1,10,35,60,35,122,-875,8160,-81890,885620,(-10291986),...; ...
where coefficients [x^n] A(x)^n and [x^n] A(x)^(n-1) are
enclosed in parenthesis and equal (-1)^n*n*A075834(n+1):
[ -1,4,-21,136,-1030,8856,-84861,894928,-10291986,128165720,...];
compare to A075834:
[1,1,1,2,7,34,206,1476,12123,111866,1143554,12816572,...]
and also to the logarithmic derivative of A075834:
[1,1,4,21,136,1030,8856,84861,894928,10291986,128165720,...].

Seiichi Manyama, Table of n, a(n) for n = 0..449

Cf. A003319, A075834, A159311, variant: A158883.

b[0] = 0; b[n_] := b[n] = n!-Sum[k!*b[n-k], {k, 1, n-1}]; a[0] = 1; a[n_] := (-1)^(n+1)*b[n]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Mar 07 2014, from 2nd formula *)

G(n,k):=(if n=k then 1 else if k=1 then (-sum(binomial(n-1,k-1)*G(n,k),k,2,n)) else sum(G(i+1,1)*G(n-i-1,k-1),i,0,n-k));
makelist(G(n,1),n,1,10); /* Vladimir Kruchinin, Mar 07 2014 */

a(n)=polcoeff(1/sum(k=0,n,(-1)^k*k!*x^k +x*O(x^n)),n)

{a(n)=local(A=[1,1]);for(i=2,n,A=concat(A,0);A[ #A]=(Vec(Ser(A)^(#A-2))-Vec(Ser(A)^(#A-1)))[ #A]);A[n+1]}

a(n) = (2-n) * a(n-1) - Sum_{k=1..n-1} a(k) * a(n-k) if n>1. - Michael Somos, Jul 23 2011
a(n) = (-1)^(n-1)*A003319(n) for n>=1.
G.f.: A(x) = 1/[Sum_{n>=0} (-1)^n*n!*x^n].
G.f. satisfies: [x^(n+1)] A(x)^n = (-1)^n*n*A075834(n+1) for n>=0.
From Sergei N. Gladkovskii, Jun 24 2012 to May 26 2013: (Start)
Continued fractions:
Let A(x) be the g.f., then A(x) = 1-x/U(0), where U(k) = x-1+x*k+(k+2)*x/U(k+1).
A(x) = 1/U(0), where U(k) = 1 - x*(2*k+1)/(1 - 2*x*(k+1)/(2*x*(k+1)- 1/U(k+1))).
G.f.: U(0), where U(k)=  1 + x*(k+1)/(1 + x*(k+1)/U(k+1)).
G.f.: 2/(G(0) + 1), where G(k)= 1 - x*(k+1)/(1 - 1/(1 + 1/G(k+1))).
G.f.: x*G(0), where G(k)=1/x + 2*k + 1 - (k+1)^2/G(k+1).
G.f.: 2/G(0), where G(k)= 1 + 1/(1 - x*(k+1)/(x*(k+1) - 1/G(k+1))). (End)

A261214 Coefficients in an asymptotic expansion of sequence A259472.

Original entry on oeis.org

1, -3, 0, -4, -33, -283, -2785, -31291, -395360, -5544754, -85427259, -1433955817, -26046643595, -509070113635, -10653941722236, -237754202827284, -5636787946661521, -141514316248243499, -3751121064314067653, -104704135027419849139, -3070176356776990397500

Offset: 0

Vaclav Kotesovec, Aug 12 2015

sign

			A259472(n)/(-2*n!) ~ 1 - 3/n - 4/n^3 - 33/n^4 - 283/n^5 - 2785/n^6 - ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..110

Cf. A003319, A260503, A259472, A261239, A261253, A261254.

Flatten[{1, Table[Sum[CoefficientList[Assuming[Element[x, Reals], Series[E^(3/x)*x^3/ExpIntegralEi[1/x]^3, {x, 0, 25}]], x][[k+1]] * StirlingS2[n-1, k-1], {k, 1, n}], {n, 1, 20}]}]

a(k) ~ -3 * k! / (4 * (log(2))^(k+1)).

For n>0, a(n) = Sum_{k=1..n} A261239(k) * Stirling2(n-1, k-1).

A261239 Coefficients in an asymptotic expansion of A259472 in falling factorials.

Original entry on oeis.org

1, -3, 0, -4, -21, -129, -910, -7242, -64155, -626319, -6685548, -77527104, -971315713, -13084909917, -188723009274, -2902997766470, -47458671376503, -821951603042523, -15037432614035864, -289828080356525052, -5870642802374608509, -124691017072423632777

Offset: 0

Vaclav Kotesovec, Aug 12 2015

sign

			A259472(n)/(-2*n!) ~ 1 - 3/n - 4/(n*(n-1)*(n-2)) - 21/(n*(n-1)*(n-2)*(n-3)) - 129/(n*(n-1)*(n-2)*(n-3)*(n-4)) - ... [coefficients are A261239]
A259472(n)/(-2*n!) ~ 1 - 3/n - 4/n^3 - 33/n^4 - 283/n^5 - 2785/n^6 - ... [coefficients are A261214]

Vaclav Kotesovec, Table of n, a(n) for n = 0..446

Cf. A003319, A260503, A259472, A261214, A261253, A261254.

CoefficientList[Assuming[Element[x, Reals], Series[E^(3/x) * x^3 / ExpIntegralEi[1/x]^3, {x, 0, 25}]], x]

a(n) ~ -3 * n! * (1 - 4/n + 2/n^2 - 2/n^3 - 31/n^4 - 288/n^5 - 2939/n^6 - 33944/n^7 - 438614/n^8 - 6266312/n^9 - 98050303/n^10), coefficients are A261253.

For n>0, a(n) = Sum_{k=1..n} A261214(k) * Stirling1(n-1, k-1).

A261253 Coefficients in an asymptotic expansion of sequence A261239.

Original entry on oeis.org

1, -4, 2, -2, -31, -288, -2939, -33944, -438614, -6266312, -98050303, -1667563622, -30631857759, -604518210964, -12758658946466, -286833669370926, -6844757550430019, -172833310268551740, -4604828067485736507, -129123684195177403168, -3801830662346341617586

Offset: 0

Vaclav Kotesovec, Aug 12 2015

sign

			A261239(n)/(-3*n!) ~ 1 - 4/n + 2/n^2 - 2/n^3 - 31/n^4 - 288/n^5 - 2939/n^6 - ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..400

Cf. A003319, A260503, A259472, A261214, A261239, A261254.

Flatten[{1, Table[Sum[CoefficientList[Assuming[Element[x, Reals], Series[E^(4/x)*x^4/ExpIntegralEi[1/x]^4, {x, 0, 25}]], x][[k+1]] * StirlingS2[n-1, k-1], {k, 1, n}], {n, 1, 25}]}]

a(k) ~ -k! / (log(2))^(k+1).

For n>0, a(n) = Sum_{k=1..n} A261254(k) * Stirling2(n-1, k-1).

A261254 Coefficients in an asymptotic expansion of A261239 in falling factorials.

Original entry on oeis.org

1, -4, 2, -4, -21, -136, -996, -8152, -73811, -733244, -7938186, -93126716, -1178054657, -15998857056, -232339375664, -3594982133808, -59070662442383, -1027605845674036, -18873206761567638, -365015243426704372, -7416392564276075453, -157957992952546414328

Offset: 0

Vaclav Kotesovec, Aug 12 2015

sign

			A261239(n)/(-3*n!) ~ 1 - 4/n + 2/(n*(n-1)) - 4/(n*(n-1)*(n-2)) - 21/(n*(n-1)*(n-2)*(n-3)) - 136/(n*(n-1)*(n-2)*(n-3)*(n-4)) - 996/(n*(n-1)*(n-2)*(n-3)*(n-4)*(n-5)) - ... [coefficients are A261254]
A261239(n)/(-3*n!) ~ 1 - 4/n + 2/n^2 - 2/n^3 - 31/n^4 - 288/n^5 - 2939/n^6 - ... [coefficients are A261253]

Vaclav Kotesovec, Table of n, a(n) for n = 0..446

Cf. A003319, A260503, A259472, A261214, A261239, A261253.

CoefficientList[Assuming[Element[x, Reals], Series[E^(4/x) * x^4 / ExpIntegralEi[1/x]^4, {x, 0, 25}]], x]

a(n) ~ -4 * n! * (1 - 5/n + 5/n^2 - 30/n^4 - 286/n^5 - 2960/n^6 - 34890/n^7 - 459705/n^8 - 6678641/n^9 - 105999991/n^10).

For n>0, a(n) = Sum_{k=1..n} A261253(k) * Stirling1(n-1, k-1).

A049295 Number of subgroups of index 4 in free group of rank n+1.

Original entry on oeis.org

1, 71, 2143, 54335, 1321471, 31817471, 764217343, 18344733695, 440294408191, 10567189327871, 253613279903743, 6086723113107455, 146081381003558911, 3505953301484470271, 84142880178680889343, 2019429129941297135615, 48466299152487396933631

Offset: 0

Valery A. Liskovets

P. de la Harpe, Topics in Geometric Group Theory, Univ. Chicago Press, 2000, p. 23, N_{4,n}.
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.13(b).

M. Hall, Subgroups of finite index in free groups, Canad. J. Math., 1 (1949), 187-190.
V. A. Liskovets and A. Mednykh, Enumeration of subgroups in the fundamental groups of orientable circle bundles over surfaces, Commun. in Algebra, 28, No. 4 (2000), 1717-1738.
Index entries for linear recurrences with constant coefficients, signature (37,-368,1436,-2256,1152).

Cf. A003319, A027837, A049290-A049294.

LinearRecurrence[{37,-368,1436,-2256,1152},{1,71,2143,54335,1321471},20] (* Harvey P. Dale, Apr 14 2016 *)

a(n) = 4*24^n-4*6^n-2*4^n+4*2^n-1.

G.f.: (264*x^3+116*x^2-34*x-1) / ((x-1)*(2*x-1)*(4*x-1)*(6*x-1)*(24*x-1)). [Colin Barker, Feb 17 2013]

More terms from Carrie Westbrook (s1213407(AT)cedarville.edu)
Terms corrected by Colin Barker, May 08 2012
a(16) from Colin Barker, Feb 17 2013

cp's OEIS Frontend

A090365 Shifts 1 place left under the INVERT transform of the BINOMIAL transform of this sequence.

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A104981 Column 1 of triangle A104980; also equals column 0 of triangle A104986, which equals the matrix logarithm of A104980.

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A111534 Main diagonal of table A111528.

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PARI

Formula

A112354 Inverse Euler transform of n!. Also the number of sequences of permutations with no global descents which are Lyndon (smallest in lexicographic order of all cyclic shifts of the sequences) where the size of the sequence = sum of sizes of the permutations.

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Maple

Mathematica

Formula

A158882 G.f. A(x) satisfies: [x^n] A(x)^n = [x^n] A(x)^(n-1) for n>1 with A(0)=A'(0)=1.

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Mathematica

Maxima

PARI

PARI

Formula

A261214 Coefficients in an asymptotic expansion of sequence A259472.

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Mathematica

Formula

A261239 Coefficients in an asymptotic expansion of A259472 in falling factorials.

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Mathematica

Formula

A261253 Coefficients in an asymptotic expansion of sequence A261239.