A063019 -id:A063019 - cp's OEIS frontend

A292751 a(n) = n!*A063019(n).

0, 1, 2, 6, 24, 240, 5040, 110880, 2298240, 47900160, 1117670400, 31654022400, 1064820556800, 39404587622400, 1532420002713600, 62512065487872000, 2723498636931072000, 128376129298120704000, 6524556605528113152000, 352829667414210674688000

Offset: 0

N. J. A. Sloane, Sep 22 2017

nonn

Robert Price, Table of n, a(n) for n = 0..104
John Engbers, David Galvin, Clifford Smyth, Restricted Stirling and Lah numbers and their inverses, arXiv:1610.05803 [math.CO], 2016.

Cf. A063019.

nmax = 20; CoefficientList[InverseSeries[x - x^2 + x^3 - x^4 + O[x]^(nmax+1), x], x]*Range[0, nmax]! (* Jean-François Alcover, Jan 21 2019 *)

seq(n)=Vec(serlaplace(serreverse(x - x^2 + x^3 - x^4 + O(x*x^n))), -(n+1)) \\ Andrew Howroyd, Nov 11 2018

Offset changed to zero and more terms added by Robert Price, Oct 19 2017

A063022 Reversion of y - y^2 - y^3 - y^5.

Original entry on oeis.org

0, 1, 1, 3, 10, 39, 161, 698, 3126, 14360, 67276, 320229, 1544257, 7528577, 37044530, 183733552, 917598103, 4610484729, 23289784660, 118209987295, 602556082765, 3083273829240, 15832177371585, 81554320766310, 421320423560400, 2182395044437686, 11332298321692704

Offset: 0

Olivier Gérard, Jul 05 2001

Robert Israel, Table of n, a(n) for n = 0..1355
Index entries for reversions of series

Cf. A063019, A063023.

with(gfun):
F:= RootOf(y-y^2-y^3-y^5-x,y):
DE:=holexprtodiffeq(F,g(x)):
Rec:= diffeqtorec(DE,g(x),a(n)):
f:= rectoproc(Rec,a(n),remember):
map(f, [$0..50]); # Robert Israel, Jan 08 2019

CoefficientList[InverseSeries[Series[y - y^2 - y^3 - y^5, {y, 0, 30}], x], x]

def Reversion(gf, n=30):
    R = PowerSeriesRing(QQ, 'x', n)
    x = R.gen().O(n)
    return list(R(gf).reverse())
Reversion(x - x^2 - x^3 - x^5, 24) # Peter Luschny, Jan 08 2019

D-finite with recurrence 575*n*(n-1)*(n-2)*(n-3)*(20979233391541*n -77947280254859)*a(n) -(n-1)*(n-2)*(n-3)*(61583500097488301*n^2 -316279381660643613*n +324795527443572336)*a(n-1) -(n-2)*(n-3)*(38717301341634153*n^3 -324199735605145484*n^2 +891613204581594443*n -818427098922228360)*a(n-2) +5*(n-3)*(15150509582201525*n^4 -167218351234002005*n^3 +671920281600084880*n^2 -1156419009962856700*n +711178431524070144)*a(n-3) +5*(-11728771987556875*n^5 +177923469670928750*n^4 -1042517573106816125*n^3 +2912399220423080050*n^2 -3791544816675160464*n +1751906653132562208)*a(n-4) -125*(5*n-26)*(5*n-22)*(5*n-23)*(73773273715*n-209652025983)*(5*n-24)*a(n-5)=0. - R. J. Mathar, Mar 21 2022

A366024 Expansion of (1/x) * Series_Reversion( x(1-x)(1+x^5) ).

Original entry on oeis.org

1, 1, 2, 5, 14, 41, 125, 393, 1265, 4147, 13799, 46488, 158261, 543610, 1881730, 6557818, 22990323, 81026013, 286915275, 1020294605, 3642192301, 13047053600, 46885795710, 168979132425, 610640337099, 2212116899436, 8031940264223, 29224761233788

Offset: 0

Seiichi Manyama, Sep 26 2023

nonn

Cf. A063019, A063026, A366023.

CoefficientList[InverseSeries[Series[x(1-x)(1+x^5),{x,0,28}],x]/x,x]  (* Stefano Spezia, Sep 26 2023 *)

a(n) = sum(k=0, n\5, (-1)^k*binomial(n+k, n)*binomial(2*n-5*k, n))/(n+1);

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/5)} (-1)^k * binomial(n+k,n) * binomial(2*n-5*k,n).

A063023 Reversion of y - y^2 - y^4 - y^5.

Original entry on oeis.org

0, 1, 1, 2, 6, 21, 77, 292, 1143, 4592, 18821, 78364, 330512, 1409149, 6063526, 26298592, 114849110, 504595293, 2228824203, 9891723114, 44087704836, 197255893945, 885630834120, 3988872011820, 18017892014655

Offset: 0

Olivier Gérard, Jul 05 2001

Robert Israel, Table of n, a(n) for n = 0..1471
Vladimir Kruchinin, The method for obtaining expressions for coefficients of reverse generating functions, arXiv:1211.3244 [math.CO], 2012.
Index entries for reversions of series

Cf. A063019, A063022.

with(gfun):
F:= RootOf(y-y^2-y^4-y^5-x, y):
DE:=holexprtodiffeq(F, g(x)):
Rec:= diffeqtorec(DE, g(x), a(n)):
f:= rectoproc(Rec, a(n), remember):
map(f, [$0..50]);# Robert Israel, Jan 08 2019

CoefficientList[InverseSeries[Series[y - y^2 - y^4 - y^5, {y, 0, 30}], x], x]

a(n):=sum((sum(binomial(j,n-k-2*j-1)*binomial(k,j),j,floor((n-k-1)/3),floor((n-k-1)/2)))*binomial(n+k-1,n-1),k,0,n-1)/n; /* Vladimir Kruchinin, May 26 2011 */

concat(0, Vec(serreverse(x - x^2 - x^4 - x^5 + O(x^30)))) \\ Michel Marcus, Jan 08 2019

# uses[Reversion from A063022]
Reversion(x - x^2 - x^4 - x^5, 25) # Peter Luschny, Jan 08 2019

a(n) = sum(k=0..n-1, (sum(j=floor((n-k-1)/3)..floor((n-k-1)/2), binomial(j,n-k-2*j-1)*binomial(k,j)))*binomial(n+k-1,n-1))/n, n>0, a(0)=0. - Vladimir Kruchinin, May 26 2011

D-finite with recurrence 500*n*(n-1)*(n-2)*(n-3)*(2749734646741*n -12465511712206)*a(n) -4350*(n-1)*(n-2)*(n-3)*(1671932639004*n^2 -9463993359665*n +8542758180998)*a(n-1) +6*(n-2)*(n-3)*(920566361953318*n^3 -5234275380165135*n^2 +1599947907526062*n + 14550951937253720)*a(n-2) -58*(n-3)*(141125952976394*n^4 -1372074713192783*n^3 +3865929553111591*n^2 -1368848933612182*n -5017756051156800)*a(n-3) +3*(-5087969954630151*n^5 +92718860230184360*n^4 -673277365411431865*n^3 +2437688480550464340*n^2 -4405429911017929324*n +3182000337895328880)*a(n-4) -145*(5*n-22)*(5*n-23)*(5*n-26)*(48367137647*n -140254765035)*(5*n-24)*a(n-5)=0. - R. J. Mathar, Mar 21 2022

A063026 Reversion of y - y^2 + y^4 - y^5.

Original entry on oeis.org

0, 1, 1, 2, 4, 9, 21, 52, 135, 368, 1045, 3068, 9230, 28245, 87414, 272544, 854012, 2685897, 8473107, 26805994, 85045674, 270599945, 863529480, 2763745020, 8870777955, 28550721966, 92128996782, 298004209496, 966085311052

Offset: 0

Olivier Gérard, Jul 05 2001

Index entries for reversions of series

Cf. A063019.

CoefficientList[InverseSeries[Series[y - y^2 + y^4 - y^5, {y, 0, 30}], x], x]

a(n+1) = (1/(n+1)) * Sum_{k=0..floor(n/3)} (-1)^k * binomial(n+k,n) * binomial(2*n-3*k,n). - Seiichi Manyama, Sep 26 2023

A364552 G.f. satisfies A(x) = 1 + xA(x) + x^4A(x)^3.

Original entry on oeis.org

1, 1, 1, 1, 2, 5, 11, 21, 39, 78, 169, 373, 808, 1727, 3719, 8153, 18100, 40315, 89770, 200250, 448755, 1010685, 2284295, 5173961, 11740697, 26699780, 60863291, 139045991, 318247190, 729572315, 1675085099, 3851795549, 8869990949, 20453679944, 47223844863

Offset: 0

Seiichi Manyama, Jul 28 2023

Cf. A000108, A071879, A346626.

Cf. A023426, A063019, A127902.

a(n) = sum(k=0, n\4, binomial(n-k, 3*k)*binomial(3*k, k)/(2*k+1));

a(n) = Sum_{k=0..floor(n/4)} binomial(n-k,3*k) * binomial(3*k,k) / (2*k+1).

A366023 Expansion of (1/x) * Series_Reversion( x(1-x)(1+x^4) ).

Original entry on oeis.org

1, 1, 2, 5, 13, 36, 104, 309, 940, 2915, 9184, 29328, 94747, 309180, 1017824, 3376693, 11279274, 37906330, 128085630, 434913555, 1483226921, 5078436800, 17450556480, 60159492600, 208013078910, 721205983737, 2506764055592, 8733076109732, 30489081691750

Offset: 0

Seiichi Manyama, Sep 26 2023

nonn

Cf. A063019, A063026, A366024.

CoefficientList[InverseSeries[Series[x(1-x)(1+x^4),{x,0,29}],x]/x,x]  (* Stefano Spezia, Sep 26 2023 *)

a(n) = sum(k=0, n\4, (-1)^k*binomial(n+k, n)*binomial(2*n-4*k, n))/(n+1);

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/4)} (-1)^k * binomial(n+k,n) * binomial(2*n-4*k,n).

A217365 Series reversion of x + x^2 + x^3 + x^4 + x^5.

Original entry on oeis.org

1, -1, 1, -1, 1, 0, -6, 27, -83, 209, -455, 845, -1169, 272, 5916, -29070, 98040, -274075, 660859, -1351756, 2110020, -1186110, -8227260, 47128770, -170898624, 505121130, -1281947030, 2772309230, -4708067030, 3936320480, 13030540120, -90168747031, 348836671587, -1077316101393

Offset: 1

R. J. Mathar, Oct 01 2012

sign

Appears to obey an 8-term hypergeometric recurrence with 4th-order polynomial coefficients.

			If y=x+x^2+x^3+x^4+x^5, then x=y -y^2 +y^3 -y^4 +y^5 -6*y^7 +27*y^8 -83*y^9 +...

R. J. Mathar and Robert Israel, Table of n, a(n) for n = 1..2066 (1..105 from R. J. Mathar)
R. J. Mathar, Series expansion of generalized Fresnel integrals, arXiv:1211.3963 (2012)

Cf. A063019 (x-x^2+x^3-x^4), A103779 (x+x^2+x^3).

rec := 5*n*(5*n-1)*(5*n+1)*(5*n+2)*(5*n+3)*a(n)-(n+1)*(27906*n^4+198109*n^3+447051*n^2+405674*n+128400)*a(n+1)+(25*(n+2))*(1875*n^4+28312*n^3+141513*n^2+287228*n+204072)*a(n+2)+(250*(n+2))*(n+3)*(250*n^3+3031*n^2+11433*n+13668)*a(n+3)+(625*(n+2))*(n+3)*(n+4)*(75*n^2+662*n+1403)*a(n+4)+(3125*(n+2))*(n+3)*(n+4)*(n+5)*(6*n+29)*a(n+5)+(3125*(n+2))*(n+3)*(n+4)*(n+5)*(n+6)*a(n+6):
f:= gfun:-rectoproc({rec,a(1) = 1, a(2) = -1, a(3) = 1, a(4) = -1, a(5) = 1, a(6) = 0},a(n),remember):
map(f, [$1..50]); # Robert Israel, Jul 10 2015

InverseSeries[x + x^2 + x^3 + x^4 + x^5 + O[x]^50][[3]] (* Vladimir Reshetnikov, Jul 09 2015 *)
RecurrenceTable[{5 (-8+3 n) (-21+5 n) (-19+5 n) (-18+5 n) (-17+5 n) (-17+6 n) (-11+6 n) a[-4+n]+4 (-3+n) (-11+6 n) (-4792620+7647427 n-4855116 n^2+1533029 n^3-240768 n^4+15048 n^5) a[-3+n]+30 (-3+n) (-2+n) (-1267420+2386123 n-1760222 n^2+636639 n^3-113004 n^4+7884 n^5) a[-2+n]+100 (-3+n) (-2+n) (-1+n) (-23+6 n) (-2310+2921 n-1152 n^2+144 n^3) a[-1+n]+125 (-3+n) (-2+n) (-1+n) n (-11+3 n) (-23+6 n) (-17+6 n) a[n]==0, a[1]==1, a[2]==-1, a[3]==1, a[4]==-1}, a, {n, 1, 40}] (* Vaclav Kotesovec, Aug 18 2015 *)

Vec(serreverse(x + x^2 + x^3 + x^4 + x^5 + O(x^50))) \\ Michel Marcus, Aug 03 2015

D-finite with recurrence: 5 n (5 n - 1) (5 n + 1) (5 n + 2) (5 n + 3) a(n) - (n + 1) (27906 n^4 + 198109 n^3 + 447051 n^2 + 405674 n + 128400) a(n + 1) + 25 (n + 2) (1875 n^4 + 28312 n^3 + 141513 n^2 + 287228 n + 204072) a(n + 2) + 250 (n + 2) (n + 3) (250 n^3 + 3031 n^2 + 11433 n + 13668) a(n + 3) + 625 (n + 2) (n + 3) (n + 4) (75 n^2 + 662 n + 1403) a(n + 4) + 3125 (n + 2) (n + 3) (n + 4) (n + 5) (6 n + 29) a(n + 5) + 3125(n + 2) (n + 3) (n + 4) (n + 5) (n + 6) a(n + 6) = 0. - Vladimir Reshetnikov, Jul 09 2015
From Robert Israel, Jul 10 2015: (Start)
G.f. G(x) satisfies G + G^2 + G^3 + G^4 + G^5 = x
and the differential equation
-2184*x^3+32760*x^2-163800*x+273000+(-10920*x^3+163800*x^2-819000*x+1365000)*G(x)+(2457000*x^4-24555336*x^3+33314040*x^2-22930200*x-10608000)*G'(x)+(11602500*x^5-88671024*x^4+64015500*x^3-18674400*x^2-27414000*x-9780000)*G''(x)+(7962500*x^6-48147528*x^5+12768480*x^4-2457200*x^3-7171500*x^2-6885000*x-1975000)*G'''(x)+(1137500*x^7-5485909*x^6-750720*x^5-1121525*x^4-654500*x^3-744375*x^2-475000*x-109375)*G''''(x) = 0
from which we can obtain the 8-term recurrence mentioned in the Comments:
1820*(5*n+1)*(5*n+2)*(5*n+3)*(5*n-1)*a(n)-13*(421993*n^4+2859670*n^3+6398855*n^2+5850050*n+1876272)*a(n+1)-60*(12512*n^4-187784*n^3-1717861*n^2-4206649*n-3230668)*a(n+2)-25*(44861*n^4+367454*n^3+1830175*n^2+6002422*n+7768608)*a(n+3)-500*(n+4)*(1309*n^3+22197*n^2+140942*n+279612)*a(n+4)-1875*(n+5)*(n+4)*(397*n^2+5657*n+18614)*a(n+5)-25000*(19*n+136)*(n+6)*(n+5)*(n+4)*a(n+6)-109375*(n+5)*(n+4)*(n+7)*(n+6)*a(n+7) = 0.
From the Lagrange inversion theorem,
  a(n) = 1/n! * (d/dx)^(n-1) (p^n)(0) where p(x) = 1/(1+x+x^2+x^3+x^4).
(End)
Recurrence: 125*(n-3)*(n-2)*(n-1)*n*(3*n - 11)*(6*n - 23)*(6*n - 17)*a(n) = -100*(n-3)*(n-2)*(n-1)*(6*n - 23)*(144*n^3 - 1152*n^2 + 2921*n - 2310)*a(n-1) - 30*(n-3)*(n-2)*(7884*n^5 - 113004*n^4 + 636639*n^3 - 1760222*n^2 + 2386123*n - 1267420)*a(n-2) - 4*(n-3)*(6*n - 11)*(15048*n^5 - 240768*n^4 + 1533029*n^3 - 4855116*n^2 + 7647427*n - 4792620)*a(n-3) - 5*(3*n - 8)*(5*n - 21)*(5*n - 19)*(5*n - 18)*(5*n - 17)*(6*n - 17)*(6*n - 11)*a(n-4). - Vaclav Kotesovec, Aug 18 2015

A364539 G.f. satisfies A(x) = 1 + xA(x) + x^3A(x)^5.

Original entry on oeis.org

1, 1, 1, 2, 7, 22, 62, 182, 583, 1928, 6358, 21063, 70888, 241889, 831634, 2874584, 9995579, 34966279, 122938956, 434062141, 1538378816, 5471697241, 19525345791, 69880082323, 250767909528, 902123110483, 3252793321513, 11753570922933, 42553831219830

Offset: 0

Seiichi Manyama, Jul 28 2023

Cf. A063019, A182454, A349311, A364522.

Cf. A023431, A071879, A215340.

a(n) = sum(k=0, n\3, binomial(n+2*k, 5*k)*binomial(5*k, k)/(4*k+1));

a(n) = Sum_{k=0..floor(n/3)} binomial(n+2*k,5*k) * binomial(5*k,k) / (4*k+1).

A365756 G.f. satisfies A(x) = 1 + xA(x) / (1 - x^3A(x)^4).

Original entry on oeis.org

1, 1, 1, 1, 2, 7, 22, 58, 142, 363, 1014, 2966, 8645, 24824, 71189, 206742, 609159, 1809493, 5388804, 16073002, 48092377, 144532884, 436168716, 1320372837, 4006489208, 12183544414, 37132838866, 113426618425, 347191793705, 1064688271730, 3270387354434

Offset: 0

Seiichi Manyama, Sep 18 2023

nonn

Cf. A023432, A212383, A365243, A365757.

Cf. A063019.

a(n) = sum(k=0, n\3, binomial(n-2*k-1, k)*binomial(n+k+1, n-3*k)/(n+k+1));

a(n) = Sum_{k=0..floor(n/3)} binomial(n-2*k-1,k) * binomial(n+k+1,n-3*k) / (n+k+1).

cp's OEIS Frontend

A292751 a(n) = n!*A063019(n).

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A063022 Reversion of y - y^2 - y^3 - y^5.

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A366024 Expansion of (1/x) * Series_Reversion( x*(1-x)*(1+x^5) ).

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A063023 Reversion of y - y^2 - y^4 - y^5.

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Maxima

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A063026 Reversion of y - y^2 + y^4 - y^5.

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A364552 G.f. satisfies A(x) = 1 + x*A(x) + x^4*A(x)^3.

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A366023 Expansion of (1/x) * Series_Reversion( x*(1-x)*(1+x^4) ).

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A217365 Series reversion of x + x^2 + x^3 + x^4 + x^5.

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A366024 Expansion of (1/x) * Series_Reversion( x(1-x)(1+x^5) ).

A364552 G.f. satisfies A(x) = 1 + xA(x) + x^4A(x)^3.

A366023 Expansion of (1/x) * Series_Reversion( x(1-x)(1+x^4) ).

A364539 G.f. satisfies A(x) = 1 + xA(x) + x^3A(x)^5.

A365756 G.f. satisfies A(x) = 1 + xA(x) / (1 - x^3A(x)^4).