cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 15 results. Next

A305114 O.g.f. A(x) satisfies: [x^n] exp( x*A(x) ) * (n^2 + 1 - A(x)) = 0 for n >= 0.

Original entry on oeis.org

1, 1, 5, 49, 818, 20902, 761661, 37594453, 2417371010, 196435255206, 19688467059690, 2386332383219634, 344073696796404628, 58206664284882779756, 11417574208958284857661, 2570602844156601626700525, 658420262904330345422925698, 190366426847612777404133776150, 61701278611229563703556400453662, 22281060851931816259923863246915334, 8914783049343376043121712182459906492
Offset: 0

Views

Author

Paul D. Hanna, May 26 2018

Keywords

Comments

Note: the factorial series, F(x) = Sum_{n>=0} n! * x^n, satisfies:
(1) [x^n] exp( x*F(x) ) * (n + 1 - F(x)) = 0 for n > 0,
(2) [x^n] exp( n * x*F(x) ) * (2 - F(x)) = 0 for n > 0.
It is remarkable that this sequence should consist entirely of integers.

Examples

			O.g.f.: A(x) = 1 + x + 5*x^2 + 49*x^3 + 818*x^4 + 20902*x^5 + 761661*x^6 + 37594453*x^7 + 2417371010*x^8 + 196435255206*x^9 + 19688467059690*x^10 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp( x*A(x) ) * (n^2 + 1 - A(x)) begins:
n=0: [0, -1, -12, -333, -21136, -2625525, -565374636, ...];
n=1: [1, 0, -9, -296, -19815, -2520504, -549669785, ...];
n=2: [4, 3, 0, -185, -15852, -2205441, -502555232, ...];
n=3: [9, 8, 15, 0, -9247, -1680336, -424030977, ...];
n=4: [16, 15, 36, 259, 0, -945189, -314097020, ...];
n=5: [25, 24, 63, 592, 11889, 0, -172753361, ...];
n=6: [36, 35, 96, 999, 26420, 1155231, 0, ...];
n=7: [49, 48, 135, 1480, 43593, 2520504, 204163063, 0, ...]; ...
in which the coefficients of x^n in row n form a diagonal of zeros.
The secondary diagonal [1, 3, 15, 259, 11889, 1155231, 204163063, ...] equals [1, 3*1, 5*3, 7*37, 9*1321, 11*105021, 13*15704851, 15*3951915073, ...], which equals the odd numbers multiplied by the coefficients in exp(x*A(x)) as shown below.
RELATED SERIES.
exp(x*A(x)) = 1 + x + 3*x^2/2! + 37*x^3/3! + 1321*x^4/4! + 105021*x^5/5! + 15704851*x^6/6! + 3951915073*x^7/7! + 1548047295537*x^8/8! + ... + E(n)*x^n/n! + ... where E(n) = (1/n^2) * Sum_{k=1..n} n!/(n-k)! * a(k) * E(n-k) for n > 0.

Links

Crossrefs

Cf. A305110, A305112, A305113, A305115, A305116.

Programs

  • PARI
    {a(n) = my(A=[1],m); for(i=1,n, A=concat(A,0); m=#A; A[m] = Vec( exp( x*(Ser(A)) ) * ((m-1)^2+1 - Ser(A)) )[m] ); A[n+1]}
for(n=0,20,print1(a(n),", "))
/* Routine to generate E(n) where exp(x*A(x)) = Sum_{n>=0} E(n) * x^n/n! */
{E(n) = if(n==0,1, (1/n^2) * sum(k=1,n, n!/(n-k)! * a(k) * E(n-k) ))}
for(n=0,20,print1(E(n),", "))

Formula

E(n) = (1/n^2) * Sum_{k=1..n} n!/(n-k)! * a(k) * E(n-k) for n > 0 with E(0) = 1, where exp(x*A(x)) = Sum_{n>=0} E(n) * x^n/n!.
a(n) ~ c * n!^2, where c = 1.509843562791802670961526317096279216134... - Vaclav Kotesovec, Oct 06 2020

A305116 O.g.f. A(x) satisfies: [x^n] exp( n^2 * x*A(x) ) * (n + 1 - A(x)) = 0 for n >= 0, where A(0) = 1.

Original entry on oeis.org

1, 1, 20, 918, 80032, 12042925, 2930093028, 1091180685420, 593430683068672, 453081063936151719, 469964400518950271900, 644367335619103754943450, 1141157288474505534959353440, 2559472926372019471694595185328, 7148083254588411836230809315647744, 24494543545202626717977721555958466300, 101668844348061438731562868186881235350528
Offset: 0

Views

Author

Paul D. Hanna, May 26 2018

Keywords

Comments

Note: the factorial series, F(x) = Sum_{n>=0} n! * x^n, satisfies:
(1) [x^n] exp( x*F(x) ) * (n + 1 - F(x)) = 0 for n > 0,
(2) [x^n] exp( n * x*F(x) ) * (2 - F(x)) = 0 for n > 0.
It is remarkable that this sequence should consist entirely of integers.

Examples

			O.g.f.: A(x) = 1 + x + 20*x^2 + 918*x^3 + 80032*x^4 + 12042925*x^5 + 2930093028*x^6 + 1091180685420*x^7 + 593430683068672*x^8 + 453081063936151719*x^9 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp( n^2 * x*A(x) ) * (n + 1 - A(x)) begins:
n=0: [0, -1, -40, -5508, -1920768, -1445151000, -2109666980160, ...];
n=1: [1, 0, -39, -5510, -1921491, -1445365884, -2109780457715, ...];
n=2: [2, 7, 0, -4780, -1823168, -1405023192, -2074130121472, ...];
n=3: [3, 26, 239, 0, -1391649, -1249241538, -1942417653741, ...];
n=4: [4, 63, 1080, 21916, 0, -860673816, -1637736990272, ...];
n=5: [5, 124, 3285, 101342, 4459057, 0, -1050171876535, ...];
n=6: [6, 215, 8096, 338580, 18744384, 1958675496, 0, ...];
n=7: [7, 342, 17355, 946660, 61910307, 6852230778, 1865443733743, 0, ...]; ...
in which the coefficients of x^n in row n form a diagonal of zeros.
RELATED SERIES.
exp(x*A(x)) = 1 + x + 3*x^2/2! + 127*x^3/3! + 22537*x^4/4! + 9717681*x^5/5! + 8729681611*x^6/6! + 14829069291583*x^7/7! + 44115361026430737*x^8/8! + ...

Links

Crossrefs

Cf. A305110, A305114, A305115, A304400, A317337.

Programs

  • PARI
    {a(n) = my(A=[1], m); for(i=1, n, A=concat(A, 0); m=#A; A[m] = Vec( exp( (m-1)^2*x*(Ser(A)) ) * ((m-1)+1 - Ser(A)) )[m] ); A[n+1]}
for(n=0, 20, print1(a(n), ", "))

Formula

a(n) ~ c * n!^3, where c = 13.46489329292094724950380929883219... - Vaclav Kotesovec, Oct 06 2020

A305112 O.g.f. A(x) satisfies: [x^n] exp( n * x*A(x) ) * (n + 2 - A(x)) = 0 for n > 0.

Original entry on oeis.org

1, 2, 14, 195, 4352, 140995, 6253116, 364374941, 27036713728, 2491365948867, 279309067129600, 37451344536549537, 5919866667150287400, 1089585227694809088555, 231042701351773513301428, 55921445108135379302601345, 15324373684519468726237630144, 4720359779903996985477019670883, 1623926325952537873282845134083944, 620377226042860947110418959221630037, 261811933632115596304376828903887527560
Offset: 0

Views

Author

Paul D. Hanna, May 26 2018

Keywords

Comments

Note: the factorial series, F(x) = Sum_{n>=0} n! * x^n, satisfies:
(1) [x^n] exp( n * x*F(x) ) * (2 - F(x)) = 0 for n > 0,
(2) [x^n] exp( x*F(x) ) * (n + 1 - F(x)) = 0 for n > 0.
It is remarkable that this sequence should consist entirely of integers.

Examples

			O.g.f.: A(x) = 1 + 2*x + 14*x^2 + 195*x^3 + 4352*x^4 + 140995*x^5 + 6253116*x^6 + 364374941*x^7 + 27036713728*x^8 + 2491365948867*x^9 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp( n * x*A(x) ) * (n + 2 - A(x)) begins:
n=0: [1, -2, -28, -1170, -104448, -16919400, -4502243520, ...];
n=1: [2, 0, -22, -1090, -100566, -16477948, -4412412190, ...];
n=2: [3, 4, 0, -738, -84304, -14795544, -4094683392, ...];
n=3: [4, 10, 44, 0, -52212, -11661618, -3524135580, ...];
n=4: [5, 18, 116, 1262, 0, -6827560, -2672928448, ...];
n=5: [6, 28, 222, 3210, 77582, 0, -1509882570, ...];
n=6: [7, 40, 368, 6030, 186864, 9166632, 0, ...];
n=7: [8, 54, 560, 9932, 335376, 21078266, 1896079208, 0, ...]; ....
in which the main diagonal is all zeros after the initial term, illustrating the  property that [x^n] exp( n * x*A(x) ) * (n + 2 - A(x)) = 0 for n > 0.
RELATED SERIES.
exp(x*A(x)) = 1 + x + 5*x^2/2! + 97*x^3/3! + 5089*x^4/4! + 550121*x^5/5! + 105094981*x^6/6! + 32297760265*x^7/7! + 14961684191777*x^8/8! + ...

Crossrefs

Cf. A305110, A305111, A305113, A305114.

Programs

  • PARI
    {a(n) = my(A=[1],m); for(i=1,n, A=concat(A,0); m=#A; A[m] = Vec( exp( (m-1)*x*(Ser(A)) ) * ((m-1) + 2 - Ser(A)) )[m] );A[n+1]}
for(n=0,20,print1(a(n),", "))

A305113 O.g.f. A(x) satisfies: [x^n] exp( n * x*A(x) ) * (n + 3 - A(x)) = 0 for n > 0.

Original entry on oeis.org

1, 3, 26, 429, 11140, 413575, 20732442, 1349324599, 110687183288, 11178507440925, 1363390073347730, 197621842687876673, 33583706596253821788, 6613771286589113270611, 1494257702398724584231946, 383919965020249799260370355, 111309986933239661174592643760, 36168040052169465186266712298305, 13090361918944129697134824216839154, 5248357548475200108500930014786066693, 2319429294524000405318027925169538777460
Offset: 0

Views

Author

Paul D. Hanna, May 26 2018

Keywords

Comments

Note: the factorial series, F(x) = Sum_{n>=0} n! * x^n, satisfies:
(1) [x^n] exp( n * x*F(x) ) * (2 - F(x)) = 0 for n > 0,
(2) [x^n] exp( x*F(x) ) * (n + 1 - F(x)) = 0 for n > 0.
It is remarkable that this sequence should consist entirely of integers.

Examples

			O.g.f.: A(x) = 1 + 3*x + 26*x^2 + 429*x^3 + 11140*x^4 + 413575*x^5 + 20732442*x^6 + 1349324599*x^7 + 110687183288*x^8 + 11178507440925*x^9 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp( n * x*A(x) ) * (n + 3 - A(x)) begins:
n=0: [2, -3, -52, -2574, -267360, -49629000, -14927358240, ...];
n=1: [3, 0, -37, -2268, -248745, -47203552, -14374490745, ...];
n=2: [4, 5, 0, -1462, -202352, -41536632, -13142258240, ...];
n=3: [5, 12, 65, 0, -121911, -32155140, -11164894659, ...];
n=4: [6, 21, 164, 2298, 0, -18516616, -8369990496, ...];
n=5: [7, 32, 303, 5636, 172075, 0, -4677722165, ...];
n=6: [8, 45, 488, 10242, 404400, 24104328, 0, ...];
n=7: [9, 60, 725, 16368, 708573, 54605228, 5760470145, 0, ...]; ...
in which the main diagonal is all zeros after the initial term, illustrating the property that [x^n] exp( n * x*A(x) ) * (n + 3 - A(x)) = 0 for n > 0.
RELATED SERIES.
exp(x*A(x)) = 1 + x + 7*x^2/2! + 175*x^3/3! + 11065*x^4/4! + 1399801*x^5/5! + 307183471*x^6/6! + 106838020087*x^7/7! + 55316481920785*x^8/8! + ...

Crossrefs

Cf. A305110, A305111, A305112, A305114.

Programs

  • PARI
    {a(n) = my(A=[1],m); for(i=1,n, A=concat(A,0); m=#A; A[m] = Vec( exp( (m-1)*x*(Ser(A)) ) * ((m-1) + 2 - Ser(A)) )[m] );A[n+1]}
for(n=0,20,print1(a(n),", "))

A305115 O.g.f. A(x) satisfies: [x^n] exp( n * x*A(x) ) * (n^2 + 1 - A(x)) = 0 for n >= 0.

Original entry on oeis.org

1, 1, 14, 450, 31144, 4041775, 890769366, 309205147860, 159530833094816, 116905524905145753, 117339344873068964150, 156605173710780053035502, 271173392660354548224099528, 596723380510396302812115056135, 1639486267597614501043345413095854, 5538914776834654404464150449671117000, 22706307619073102796968257487359193429120
Offset: 0

Views

Author

Paul D. Hanna, May 26 2018

Keywords

Comments

Note: the factorial series, F(x) = Sum_{n>=0} n! * x^n, satisfies:
(1) [x^n] exp( n * x*F(x) ) * (2 - F(x)) = 0 for n > 0,
(2) [x^n] exp( x*F(x) ) * (n + 1 - F(x)) = 0 for n > 0.
It is remarkable that this sequence should consist entirely of integers.

Examples

			O.g.f.: A(x) = 1 + x + 14*x^2 + 450*x^3 + 31144*x^4 + 4041775*x^5 + 890769366*x^6 + 309205147860*x^7 + 159530833094816*x^8 + 116905524905145753*x^9 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp( n * x*A(x) ) * (n^2 + 1 - A(x)) begins:
n=0: [0, -1, -28, -2700, -747456, -485013000, -641353943520, ...];
n=1: [1, 0, -27, -2702, -747963, -485118684, -641396951615, ...];
n=2: [4, 7, 0, -2092, -678784, -462055752, -623679177536, ...];
n=3: [9, 26, 101, 0, -460275, -391250658, -569892209247, ...];
n=4: [16, 63, 348, 4828, 0, -246538056, -461135488928, ...];
n=5: [25, 124, 837, 14150, 810509, 0, -277891671695, ...];
n=6: [36, 215, 1688, 30348, 2099712, 378224376, 0, ...];
n=7: [49, 342, 3045, 56548, 4020741, 920163738, 393372598609, 0, ...]; ...
in which the coefficients of x^n in row n form a diagonal of zeros.
RELATED SERIES.
exp(x*A(x)) = 1 + x + 3*x^2/2! + 91*x^3/3! + 11161*x^4/4! + 3793881*x^5/5! + 2933070331*x^6/6! + 4510118566003*x^7/7! + 12503335235913201*x^8/8! + ...

Links

Crossrefs

Cf. A305110, A305114, A305116, A317337.

Programs

  • PARI
    {a(n) = my(A=[1], m); for(i=1, n, A=concat(A, 0); m=#A; A[m] = Vec( exp( (m-1)*x*(Ser(A)) ) * ((m-1)^2+1 - Ser(A)) )[m] ); A[n+1]}
for(n=0, 20, print1(a(n), ", "))

Formula

a(n) ~ c * n!^3, where c = 2.49393609789981559563078907122202821077556480458411... - Vaclav Kotesovec, Oct 06 2020

A305111 O.g.f. A(x) satisfies: [x^n] exp( n * x*A(x) ) * (n - A(x)) = 0 for n >= 0.

Original entry on oeis.org

1, 0, 2, 15, 232, 5335, 175416, 7847665, 460083056, 34295632587, 3171572232860, 356568882856505, 47917706485943520, 7587855756759594511, 1398620339344282321232, 296926334953470628671285, 71938929865159851175220384, 19730017662046372201511846043, 6081674963212598041745828898948, 2093499393528764015996786711595373, 800173558940733992481330233586267600
Offset: 0

Views

Author

Paul D. Hanna, May 26 2018

Keywords

Comments

Note: the factorial series, F(x) = Sum_{n>=0} n! * x^n, satisfies:
(1) [x^n] exp( n * x*F(x) ) * (2 - F(x)) = 0 for n > 0,
(2) [x^n] exp( x*F(x) ) * (n + 1 - F(x)) = 0 for n > 0.
It is remarkable that this sequence should consist entirely of integers.

Examples

			O.g.f.: A(x) = 1 + 2*x^2 + 15*x^3 + 232*x^4 + 5335*x^5 + 175416*x^6 + 7847665*x^7 + 460083056*x^8 + 34295632587*x^9 + 3171572232860*x^10 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp( n * x*A(x) ) * (n - A(x)) begins:
n=0: [-1, 0, -4, -90, -5568, -640200, -126299520, ...];
n=1: [0, 0, -4, -102, -5952, -669460, -130272180, ...];
n=2: [1, 2, 0, -82, -5456, -636888, -126025856, ...];
n=3: [2, 6, 14, 0, -3678, -527934, -112372002, ...];
n=4: [3, 12, 44, 198, 0, -323848, -87935232, ...];
n=5: [4, 20, 96, 590, 6532, 0, -51109520, ...];
n=6: [5, 30, 176, 1278, 17328, 476520, 0, ...];
n=7: [6, 42, 290, 2388, 34374, 1150022, 67654674, 0, ...];
in which the main diagonal is all zeros after the initial term, illustrating the property that exp( n * x*A(x) ) * (n - A(x)) for n >= 0.

Crossrefs

Cf. A305110, A305112, A305113, A305114.

Programs

  • PARI
    {a(n) = my(A=[1,0],m); for(i=1,n, A=concat(A,0); m=#A; A[m] = Vec( exp( (m-1)*x*(Ser(A)) ) * (m-1 - Ser(A)) )[m] );A[n+1]}
for(n=0,20,print1(a(n),", "))

A317337 O.g.f. A(x) satisfies: [x^n] exp( n^3*x*A(x) ) * (n+1 - n*A(x)) = 0 for n>=1.

Original entry on oeis.org

1, 1, 12, 729, 111440, 31377625, 14001201036, 9064452341847, 8027821828474816, 9322437359885669613, 13746212321035446900300, 25094943743950232692612534, 55574014665416527079564569056, 146797467684802516650481763597455, 456012687037844090869850529901126900, 1645914373011657806464530612985244787000
Offset: 0

Views

Author

Paul D. Hanna, Aug 01 2018

Keywords

Comments

Compare: the factorial series F(x) = Sum_{n>=0} n!*x^n satisfies
(1) [x^n] exp( n^2*x*F(x) ) * (n + 1 - n*F(x)) = 0 for n>=1,
(2) [x^n] exp( n^3*x*F(x) ) * (n^2 + 1 - n^2*F(x)) = 0 for n>=1.
It is remarkable that this sequence should consist entirely of integers.

Examples

			O.g.f.: A(x) = 1 + x + 12*x^2 + 729*x^3 + 111440*x^4 + 31377625*x^5 + 14001201036*x^6 + 9064452341847*x^7 + 8027821828474816*x^8 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp( n^3*x*A(x) ) * (n+1 - n*A(x)) begins:
n=1: [1, 0, -23, -4376, -2674995, -3765464504, ...];
n=2: [1, 6, 0, -8908, -5494464, -7640806512, ...];
n=3: [1, 24, 549, 0, -8632395, -12056269968, ...];
n=4: [1, 60, 3616, 204712, 0, -17114998496, -45010750350080, ...];
n=5: [1, 120, 14505, 1750880, 197597325, 0, -60559334101475, ...];
n=6: [1, 210, 44352, 9406044, 1987128000, 391935493296, 0, ...]; ...
in which the coefficients of x^n in row n form a diagonal of zeros.

Links

Crossrefs

Cf. A305110, A305114, A305115, A305116, A317338.

Programs

  • PARI
    {a(n) = my(A=[1], m); for(i=1, n, A=concat(A, 0); m=#A; A[m] = Vec( exp( (m-1)^3*x*(Ser(A)) ) * (m - (m-1)*Ser(A)) )[m]/(m-1) ); A[n+1]}
for(n=0, 20, print1(a(n), ", "))

Formula

a(n) ~ sqrt(1-c) * 3^(3*n) * n^(2*n - 3/2) / (sqrt(2*Pi) * c^n * (3-c)^(2*n) * exp(2*n)), where c = -LambertW(-3*exp(-3)) = 0.1785606278779211065968086697... = -A226750. - Vaclav Kotesovec, Aug 06 2018

A305602 G.f. A(x) satisfies: A(x) = 1 + x*[d/dx 1/(1 - x*A(x)^2)].

Original entry on oeis.org

1, 1, 6, 54, 628, 8760, 140904, 2552151, 51243864, 1127982321, 26993774100, 697703846499, 19372450060296, 575205186725962, 18191422973198622, 610655961723782310, 21689599103526363600, 812832263931582168447, 32057155649057309677062, 1327393477257351399000744, 57581802198755959140129600
Offset: 0

Views

Author

Paul D. Hanna, Jun 05 2018

Keywords

Examples

			G.f.: A(x) = 1 + x + 6*x^2 + 54*x^3 + 628*x^4 + 8760*x^5 + 140904*x^6 + 2552151*x^7 + 51243864*x^8 + 1127982321*x^9 + 26993774100*x^10 + ...
such that A(x) = 1 + x*[d/dx 1/(1 - x*A(x)^2)].
RELATED SERIES.
A(x)^2 = 1 + 2*x + 13*x^2 + 120*x^3 + 1400*x^4 + 19424*x^5 + 309780*x^6 + 5559054*x^7 + 110623342*x^8 + 2415298374*x^9 + 57387784542*x^10 + ...
1/(1 - x*A(x)^2) = 1 + x + 3*x^2 + 18*x^3 + 157*x^4 + 1752*x^5 + 23484*x^6 + 364593*x^7 + 6405483*x^8 + 125331369*x^9 + 2699377410*x^10 + ...
exp( Integral A(x)^2 dx ) = 1 + x + 3*x^2/2! + 33*x^3/3! + 849*x^4/4! + 38061*x^5/5! + 2575611*x^6/6! + 242377533*x^7/7! + 30085188993*x^8/8! + ...
A'(x)/A(x) = 1 + 11*x + 145*x^2 + 2247*x^3 + 39461*x^4 + 768983*x^5 + 16409646*x^6 + 380013063*x^7 + 9487631035*x^8 + 254076973011*x^9 + ...

Links

Crossrefs

Cf. A305603, A305604, A209881, A305110.

Programs

  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A=1+x*deriv(1/(1-x*A^2+x*O(x^n)))); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
  • PARI
    {a(n) = my(A=[1], m); for(i=1, n, A=concat(A, 0); m=#A; A[m] = Vec( exp( (m-1)*intformal(Ser(A)^2) ) * ((m-1) + 1 - Ser(A)) )[m] ); A[n+1]}
for(n=0, 25, print1(a(n), ", "))

Formula

O.g.f. A(x) satisfies:
(1) [x^n] exp( n * Integral A(x)^2 dx ) * (n + 1 - A(x)) = 0 for n > 0.
(2) A(x) = 1 + x*A(x)*(A(x) + 2*x*A'(x))/(1 - x*A(x)^2)^2.
a(n) ~ c * 2^n * n^(3/2) * n!, where c = 0.26934871195193907483980578... - Vaclav Kotesovec, Oct 06 2020

A305603 G.f. A(x) satisfies: A(x) = 1 + x*[d/dx 1/(1 - x*A(x)^3)].

Original entry on oeis.org

1, 1, 8, 102, 1712, 34785, 819384, 21810124, 645122272, 20957720148, 741260263600, 28350052179438, 1165931175542064, 51320048879474206, 2407857124657086480, 119990501174741855400, 6330579163195128292800, 352584892981590315935084, 20675941712941698695206368, 1273517057922072215818491064, 82210136955409063394289646720
Offset: 0

Views

Author

Paul D. Hanna, Jun 05 2018

Keywords

Examples

			G.f.: A(x) = 1 + x + 8*x^2 + 102*x^3 + 1712*x^4 + 34785*x^5 + 819384*x^6 + 21810124*x^7 + 645122272*x^8 + 20957720148*x^9 + 741260263600*x^10 + ...
such that A(x) = 1 + x*[d/dx 1/(1 - x*A(x)^3)].
RELATED SERIES.
A(x)^3 = 1 + 3*x + 27*x^2 + 355*x^3 + 5964*x^4 + 120021*x^5 + 2790794*x^6 + 73301427*x^7 + 2141393220*x^8 + 68800518492*x^9 + ...
1/(1 - x*A(x)^3) = 1 + x + 4*x^2 + 34*x^3 + 428*x^4 + 6957*x^5 + 136564*x^6 + 3115732*x^7 + 80640284*x^8 + 2328635572*x^9 + ...
A'(x)/A(x) = 1 + 15*x + 283*x^2 + 6343*x^3 + 162076*x^4 + 4614153*x^5 + 144287466*x^6 + 4908441479*x^7 + 180383821348*x^8 + 7122692545660*x^9 + ...

Links

Crossrefs

Cf. A305602, A305604, A209881, A305110.

Programs

  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A=1+x*deriv(1/(1-x*A^3+x*O(x^n)))); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
  • PARI
    {a(n) = my(A=[1], m); for(i=1, n, A=concat(A, 0); m=#A; A[m] = Vec( exp( (m-1)*intformal(Ser(A)^3) ) * ((m-1) + 1 - Ser(A)) )[m] ); A[n+1]}
for(n=0, 25, print1(a(n), ", "))

Formula

O.g.f. A(x) satisfies:
(1) [x^n] exp( n * Integral A(x)^3 dx ) * (n + 1 - A(x)) = 0 for n > 0.
(2) A(x) = 1 + x*A(x)^2*(A(x) + 3*x*A'(x))/(1 - x*A(x)^3)^2.
a(n) ~ c * 3^n * n^(4/3) * n!, where c = 0.1925904251831569484470022... - Vaclav Kotesovec, Oct 06 2020

A305604 G.f. A(x) satisfies: A(x) = 1 + x*[d/dx 1/(1 - x*A(x)^4)].

Original entry on oeis.org

1, 1, 10, 165, 3620, 96600, 2996586, 105222740, 4110953640, 176563668420, 8262487393200, 418357209052255, 22790291724038580, 1329373540730613885, 82689132241752729840, 5464831115387029530825, 382492359049214499350160, 28269306654212844076331775, 2200386909130514596330158960, 179936864840741390285785502825, 15424543660650808111807757322400
Offset: 0

Views

Author

Paul D. Hanna, Jun 05 2018

Keywords

Examples

			G.f.: A(x) = 1 + x + 10*x^2 + 165*x^3 + 3620*x^4 + 96600*x^5 + 2996586*x^6 + 105222740*x^7 + 4110953640*x^8 + 176563668420*x^9 + ...
such that A(x) = 1 + x*[d/dx 1/(1 - x*A(x)^4)].
RELATED SERIES.
A(x)^4 = 1 + 4*x + 46*x^2 + 784*x^3 + 17181*x^4 + 452860*x^5 + 13831594*x^6 + 478200572*x^7 + 18418253542*x^8 + 781180290204*x^9 + ...
1/(1 - x*A(x)^4) = 1 + x + 5*x^2 + 55*x^3 + 905*x^4 + 19320*x^5 + 499431*x^6 + 15031820*x^7 + 513869205*x^8 + 19618185380*x^9 + ...
A'(x)/A(x) = 1 + 19*x + 466*x^2 + 13659*x^3 + 457926*x^4 + 17142730*x^5 + 706064549*x^6 + 31677960427*x^7 + 1537022113117*x^8 + ...

Links

Crossrefs

Cf. A305602, A305603, A209881, A305110.

Programs

  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A=1+x*deriv(1/(1 - x*A^4+x*O(x^n)))); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
  • PARI
    {a(n) = my(A=[1], m); for(i=1, n, A=concat(A, 0); m=#A; A[m] = Vec( exp( (m-1)*intformal(Ser(A)^4) ) * ((m-1) + 1 - Ser(A)) )[m] ); A[n+1]}
for(n=0, 25, print1(a(n), ", "))

Formula

O.g.f. A(x) satisfies:
(1) [x^n] exp( n * Integral A(x)^4 dx ) * (n + 1 - A(x)) = 0 for n > 0.
(2) A(x) = 1 + x*A(x)^3*(A(x) + 4*x*A'(x))/(1 - x*A(x)^4)^2.
a(n) ~ c * 4^n * n^(5/4) * n!, where c = 0.147639333661398142298711... - Vaclav Kotesovec, Oct 06 2020
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