A077607 -id:A077607 - cp's OEIS frontend

A260491 Coefficients in asymptotic expansion of sequence A077607.

1, -4, 0, -8, -76, -752, -8460, -107520, -1522124, -23717424, -402941324, -7407988448, -146479479308, -3099229422352, -69863683041868, -1671667534710720, -42318672085310540, -1130167625049525232, -31758424368739424780, -936840101208573355680

Offset: 0

Vaclav Kotesovec, Jul 27 2015

sign

For k > 2 is a(k) negative.

			A077607(n) / (-n!) ~ 1 - 4/n - 8/n^3 - 76/n^4 - 752/n^5 - 8460/n^6 - ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..126
Richard J. Martin, and Michael J. Kearney, Integral representation of certain combinatorial recurrences, Combinatorica: 35:3 (2015), 309-315.

Cf. A077607, A111529, A256168, A260503, A260530, A260532, A260578.

nmax = 30; b = CoefficientList[Assuming[Element[x, Reals], Series[x^4*E^(2/x)/(ExpIntegralEi[1/x] - x*E^(1/x))^2, {x, 0, nmax}]], x]; Flatten[{1, Table[Sum[b[[k+1]]*StirlingS2[n-1, k-1], {k, 1, n}], {n, 1, nmax}]}] (* Vaclav Kotesovec, Aug 03 2015 *)

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

A078140 Convolutory inverse of signed lower Wythoff sequence.

Original entry on oeis.org

1, 3, 5, 9, 17, 30, 52, 90, 154, 262, 446, 758, 1285, 2176, 3683, 6230, 10533, 17803, 30085, 50831, 85873, 145063, 245037, 413891, 699082, 1180761, 1994293, 3368302, 5688920, 9608292, 16227841, 27407792, 46289925, 78180465, 132041227

Offset: 1

Clark Kimberling, Nov 23 2002

nonn

Suppose that r is a real number in the interval [3/2, 5/3).  Let C(r) = (c(k)) be the sequence of coefficients in the Maclaurin series for 1/(Sum_{k>=0} floor((k+1)*r))(-x)^k).  It appears that c(k) > 0 for all k >= 0.  Indeed, it appears that C(r) is strictly increasing and that the limit L(r) of c(k+1)/c(k) as k -> oo exists.  Following is a guide for selected numbers r.
** r **           C(r)       L(r)
sqrt(7/3)        A188135    A288238
Pi/2             A288229    A288239
sqrt(5/2)        A288230    A288240
4^(1/3)          A288231    A288241
(1 + sqrt(5))/2  A078140    A281112
3e/5             A288232    A288242
sqrt(8/3)        A288233    A288935
-1 + sqrt(7)     A288234    A289003
sqrt(e)          A288235    A289005
-4/5 + sqrt(6)   A288236    A289032
sqrt(11/4)       A288237    A289033

			a(5) = 17 = -[w(5)*a(1)-w(4)*a(2)+w(3)*a(3)-w(2)*a(4)] = -8*1+6*3-4*5+3*9. (a(1),a(2),...,a(n))(*)(w(1),-w(2),w(3),...,-d*w(n)) = (1,0,0,...,0), where (*) denotes convolution, w = lower Wythoff sequence, A000201.

Clark Kimberling, Table of n, a(n) for n = 1..1000
Clark Kimberling, Another question about the golden ratio and other numbers, MathOverflow, Jan 17 2017.

Cf. A000201, A077607, A281112, A279676.

CoefficientList[Series[1/Sum[Floor[GoldenRatio*(k + 1)] (-x)^k, {k, 0, 50}],
{x, 0,50}], x]  (* Clark Kimberling, Dec 12 2016 *)

a(n) = d*[w(n)*a(1)-w(n-1)*a(2)+...+d*w(2)*a(n-1)], where d=(-1)^n, with a(1)=1 and w=floor(n*tau), tau=(1+sqrt(5))/2.

Comments added by Clark Kimberling, Jul 10 2017

A111529 Row 2 of table A111528.

Original entry on oeis.org

1, 1, 4, 22, 148, 1156, 10192, 99688, 1069168, 12468208, 157071424, 2126386912, 30797423680, 475378906432, 7793485765888, 135284756985472, 2479535560687360, 47860569736036096, 970606394944476160, 20635652201785613824, 459015456156148876288, 10662527360021306782720

Offset: 0

Paul D. Hanna, Aug 06 2005

			(1/2)*log(1 + 2*x + 6*x^2 + ... + ((n+1)!/1!)*x^n + ...)
= x + (4/2)*x^2 + (22/3)*x^3 + (148/4)*x^4 + (1156/5)*x^5 + ...

Robert Israel, Table of n, a(n) for n = 0..410
Paul Barry, A note on number triangles that are almost their own production matrix, arXiv:1804.06801 [math.CO], 2018.
Richard J. Martin and Michael J. Kearney, Integral representation of certain combinatorial recurrences, Combinatorica: 35:3 (2015), 309-315.
A. N. Stokes, Continued fraction solutions of the Riccati equation, Bull. Austral. Math. Soc. Vol. 25 (1982), 207-214.

Cf. A111528 (table), A003319 (row 1), A111530 (row 3), A111531 (row 4), A111532 (row 5), A111533 (row 6), A111534 (diagonal).

Cf. A077607, A089949, A260491.

N:= 30: # to get a(0) to a(N)
g:= 1/2*log(add((n+1)!*x^n,n=0..N+1)):
S:= series(g,x,N+1);
1, seq(j*coeff(S,x,j),j=0..N); # Robert Israel, Jul 10 2015

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[2, n];
Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Aug 09 2018 *)

{a(n)=if(n<0,0,if(n==0,1, (n/2)*polcoeff(log(sum(m=0,n,(m+1)!/1!*x^m)),n)))}

G.f.: (1/2)*log(Sum_{n >= 0} (n+1)!*x^n) = Sum_{n >= 1} a(n)*x^n/n.
G.f.: 1/(1+2*x - 3*x/(1+3*x - 4*x/(1+4*x - ... (continued fraction).
a(n) = Sum_{k = 0..n} 2^(n-k)*A089949(n,k). - Philippe Deléham, Oct 16 2006
G.f. 1/(2*x-G(0)) where G(k) = 2*x - 1 - k*x - x*(k+1)/G(k+1); G(0)=x (continued fraction, Euler's 1st kind, 1-step). - Sergei N. Gladkovskii, Aug 14 2012
G.f.: 1/(2*x) - 1/(G(0) - 1) where G(k) = 1 + x*(k+1)/(1 - 1/(1 + 1/G(k+1)));(continued fraction, 3-step). - Sergei N. Gladkovskii, Nov 20 2012
G.f.: 1 + x/(G(0)-2*x) where G(k) = 1 + (k+1)*x - x*(k+3)/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Dec 26 2012
G.f.: (1 + 1/Q(0))/2, where Q(k) = 1 + k*x - x*(k+2)/Q(k+1); (continued fraction). In general, the g.f. for row (r+2) is (r + 1 + 1/Q(0))/(r + 2). - Sergei N. Gladkovskii, May 04 2013
G.f.: W(0), where W(k) = 1 - x*(k+1)/( x*(k+1) - 1/(1 - x*(k+3)/( x*(k+3) - 1/W(k+1) ))); (continued fraction). - Sergei N. Gladkovskii, Aug 26 2013
a(n) ~ n! * n^2/2 * (1 - 1/n - 2/n^2 - 8/n^3 - 52/n^4 - 436/n^5 - 4404/n^6 - 51572/n^7 - 683428/n^8 - 10080068/n^9 - 163471284/n^10), where the coefficients are given by (n+2)*(n+1)/n^2 * Sum_{k>=0} A260491(k)/(n+2)^k. - Vaclav Kotesovec, Jul 27 2015
a(n) = -A077607(n+2)/2. - Vaclav Kotesovec, Jul 29 2015
From Peter Bala, Jul 12 2022: (Start)
O.g.f: A(x) = ( Sum_{k >= 0} ((k+2)!/2!)*x^k )/( Sum_{k >= 0} (k+1)!*x^k ).
A(x)/(1 - 2*x*A(x)) = Sum_{k >= 0} ((k+2)!/2!)*x^k.
Riccati differential equation: x^2*A'(x) + 2*x*A^2(x) - (1 + x)*A(x) + 1 = 0.
Apply Stokes 1982 to find that A(x) = 1/(1 - x/(1 - 3*x/(1 - 2*x/(1 - 4*x/(1 - 3*x/(1 - 5*x/(1 - ... - n*x/(1 - (n+2)*x/(1 - ...))))))))), a continued fraction of Stieltjes type. (End)

A225127 Convolutory inverse of the nonprimes.

Original entry on oeis.org

1, -4, 10, -24, 59, -146, 360, -886, 2182, -5376, 13244, -32624, 80364, -197968, 487672, -1201319, 2959297, -7289859, 17957662, -44236464, 108971015, -268436517, 661259918, -1628931424, 4012669610, -9884711639, 24349755585, -59982589144, 147759635098

Offset: 1

Clark Kimberling, Apr 29 2013

Coefficients in 1/(1+g(x)), where g is the generating functions of the sequence of nonprimes: (1,4,6,8,9,...). For the convolutory inverse of the primes, see A030018. Conjecture: a(n+1)/a(n) has a limit, -2.4633754095588889..., analogous to the Backhouse constant.

The sequences with nonzero first term form a group under convolution. The identity is (1,0,0,0,...), and the inverse of a sequence r(1), r(2), r(3), ... is s(1), s(2), s(3),... given by s(1) = 1/r(1) and s(n) = -(r(2)*s(n-1) + ... + r(n)*s(1))/r(1). Thus, s(i) are the coefficients of the power series for 1/(r(1) + r(2)*x + r(3)*x^2 + ... ).

			(1,4,6,8,9,...)**(1,-4,10,-24,59,...) = (1,0,0,0,0,...), where ** denotes convolution.

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A030018, A077607.

z = 1000; c = Complement[Range[z], Prime[Range[PrimePi[z]]]]; r[n_] := r[n] = c[[n]]; k[n_] := k[n] = 0; k[1] = 1; a[n_] := a[n] = (k[n] - Sum[r[i]*a[n - i + 1], {i, 2, n}])/r[1]; t = Table[a[n], {n, 1, 40}]   (* A225127 *)

A296617 Expansion of 1/Sum_{k>=0} (k+1)^(k+1)*x^k.

Original entry on oeis.org

1, -4, -11, -104, -1388, -22980, -446524, -9882944, -244592124, -6684031040, -199824449532, -6488250797312, -227456440349948, -8565880619584896, -345018776767586572, -14805421633750610240, -674514253891722861612, -32522567276377571337728

Offset: 0

Seiichi Manyama, Dec 19 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..385

Cf. A000312, A077607, A294372, A296715.

N=66; x='x+O('x^N); Vec(1/sum(k=0, N, (k+1)^(k+1)*x^k))

A303671 a(n) = [x^n] (6 / Sum_{k=0..n} (k+3)!*x^k)^(1/2).

Original entry on oeis.org

1, -2, -4, -20, -140, -1192, -11656, -127072, -1517480, -19624400, -272656064, -4046157472, -63844717184, -1067307017600, -18846094327360, -350578477836160, -6854465726916640, -140564170444594880, -3017526392523529600, -67690323794750627200

Offset: 0

Seiichi Manyama, Apr 28 2018

sign

Seiichi Manyama, Table of n, a(n) for n = 0..446

Cf. A077607, A303672, A303673.

[seq(coeff(series((factorial(3)/add(factorial(k+3)*x^k,k=0..n))^(1/2), x,30),x,n),n=0..25)]; # Muniru A Asiru, Apr 29 2018

N=66; x='x+O('x^N); Vec((3!/sum(k=0, N, (k+3)!*x^k))^(1/2))

A303672 a(n) = [x^n] (5040 / Sum_{k=0..n} (k+7)!*x^k)^(1/4).

Original entry on oeis.org

1, -2, -8, -60, -600, -7152, -96456, -1430544, -22933920, -393013280, -7144207808, -137001926144, -2760226854816, -58242464679360, -1283886660610560, -29506641823939200, -705788634473952000, -17544801385483584000, -452659831142363014400

Offset: 0

Seiichi Manyama, Apr 28 2018

sign

Seiichi Manyama, Table of n, a(n) for n = 0..444

Cf. A077607, A303671, A303673.

[seq(coeff(series((factorial(7)/add(factorial(k+7)*x^k,k=0..n))^(1/4), x,30),x,n),n=0..25)]; # Muniru A Asiru, Apr 29 2018

N=66; x='x+O('x^N); Vec((7!/sum(k=0, N, (k+7)!*x^k))^(1/4))

A303673 a(n) = [x^n] (15! / Sum_{k=0..n} (k+15)!*x^k)^(1/8).

Original entry on oeis.org

1, -2, -16, -204, -3264, -60384, -1239912, -27591408, -655324704, -16443029408, -432731364992, -11882079044992, -339084517212576, -10026995732141760, -306530743192692480, -9669854410016300160, -314315622535266332160

Offset: 0

Seiichi Manyama, Apr 28 2018

sign

Seiichi Manyama, Table of n, a(n) for n = 0..439

Cf. A077607, A303671, A303672.

[seq(coeff(series((factorial(15)/add(factorial(k+15)*x^k,k=0..n))^(1/8), x,30),x,n),n=0..25)]; # Muniru A Asiru, Apr 29 2018

N=66; x='x+O('x^N); Vec((15!/sum(k=0, N, (k+15)!*x^k))^(1/8))

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A260491 Coefficients in asymptotic expansion of sequence A077607.

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A078140 Convolutory inverse of signed lower Wythoff sequence.

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A111529 Row 2 of table A111528.

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A225127 Convolutory inverse of the nonprimes.

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A296617 Expansion of 1/Sum_{k>=0} (k+1)^(k+1)*x^k.

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A303671 a(n) = [x^n] (6 / Sum_{k=0..n} (k+3)!*x^k)^(1/2).

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A303672 a(n) = [x^n] (5040 / Sum_{k=0..n} (k+7)!*x^k)^(1/4).

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A303673 a(n) = [x^n] (15! / Sum_{k=0..n} (k+15)!*x^k)^(1/8).

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