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 1-5 of 5 results.

A296171 O.g.f. A(x) satisfies: [x^n] exp( n^2 * A(x) ) = [x^(n-1)] exp( n^2 * A(x) ) for n>=1.

Original entry on oeis.org

1, -1, -1, -9, -134, -2852, -79096, -2699480, -109201844, -5100872244, -269903909820, -15944040740604, -1039553309158964, -74123498185170292, -5736368141560365292, -478780244956262592748, -42865943103053965559668, -4097785410628237071311764, -416572537937169684523985420, -44873737158384968851319470220, -5106038963454360810619516396820, -611986780692307637617151164361140, -77066319756799442735378541663266476
Offset: 1

Views

Author

Paul D. Hanna, Dec 07 2017

Keywords

Comments

E.g.f. G(x) of A296170 satisfies: [x^(n-1)] G(x)^(n^2) = [x^n] G(x)^(n^2) for n>=1.

Examples

			G.f. A(x) = x - x^2 - x^3 - 9*x^4 - 134*x^5 - 2852*x^6 - 79096*x^7 - 2699480*x^8 - 109201844*x^9 - 5100872244*x^10 - 269903909820*x^11 - 15944040740604*x^12 - 1039553309158964*x^13 - 74123498185170292*x^14 - 5736368141560365292*x^15 + ...
such that
G(x) = exp(A(x)) = 1 + x - x^2/2! - 11*x^3/3! - 239*x^4/4! - 17059*x^5/5! - 2145689*x^6/6! - 412595231*x^7/7! - 111962826751*x^8/8! - 40590007936199*x^9/9! - 18900753214178609*x^10/10! + ... + A296170(n)*x^n/n! + ...
satisfies [x^(n-1)] G(x)^(n^2) = [x^n] G(x)^(n^2) for n>=1.
RELATED SERIES.
Series_Reversion(A(x)) = x + x^2 + 3*x^3 + 19*x^4 + 226*x^5 + 4259*x^6 + 110514*x^7 + 3626207*x^8 + 143043592*x^9 + 6567931068*x^10 + 343278693103*x^11 + 20092744961109*x^12 + 1300754163383700*x^13 + ... + A295812(n)*x^n + ...

Links

Crossrefs

Cf. A296170, A295812, A296173, A296175, A296177.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1,n+1, A=concat(A,0); V=Vec(Ser(A)^((#A-1)^2)); A[#A] = (V[#A-1] - V[#A])/(#A-1)^2 ); polcoeff(log(Ser(A)),n)}
for(n=1,30,print1(a(n),", "))

A296172 E.g.f. A(x) satisfies: [x^(n-1)] A(x)^(n^3) = [x^n] A(x)^(n^3) for n>=1.

Original entry on oeis.org

1, 1, -5, -197, -65111, -62390159, -125012786669, -447082993406405, -2583111044504384687, -22511408975342644804991, -281350305428215911326408789, -4850582201056517165575319399909, -111834955668396093904661955538037255, -3361788412998032560821833199260880942287, -128987969989211586699135087535153035663946301, -6203990036027464835833031041177436339788197962789
Offset: 0

Views

Author

Paul D. Hanna, Dec 07 2017

Keywords

Comments

Compare e.g.f. to: [x^(n-1)] exp(x)^n = [x^n] exp(x)^n for n>=1.

Examples

			E.g.f.: A(x) = 1 + x - 5*x^2/2! - 197*x^3/3! - 65111*x^4/4! - 62390159*x^5/5! - 125012786669*x^6/6! - 447082993406405*x^7/7! - 2583111044504384687*x^8/8! - 22511408975342644804991*x^9/9! - 281350305428215911326408789*x^10/10! - 4850582201056517165575319399909*x^11/11! - 111834955668396093904661955538037255*x^12/12! +...
To illustrate [x^(n-1)] A(x)^(n^3) = [x^n] A(x)^(n^3), form a table of coefficients of x^k in A(x)^(n^3) that begins as
n=1: [(1), (1), -5/2, -197/6, -65111/24, -62390159/120, -125012786669/720, ...];
n=2: [1, (8), (8), -1040/3, -71152/3, -64676744/15, -63817770776/45, ...];
n=3: [1, 27, (567/2), (567/2), -787941/8, -648507951/40, -405807483249/80, ...];
n=4: [1, 64, 1856, (88448/3), (88448/3), -689015872/15, -611019817664/45, ...];
n=5: [1, 125, 14875/2, 1649375/6, (156207625/24), (156207625/24), ...];
n=6: [1, 216, 22680, 1533168, 73812816, (12455715384/5), (12455715384/5), ...];
n=7: [1, 343, 115591/2, 38174185/6, 12294445009/24, 3808296195823/120, (1051338418817239/720), (1051338418817239/720), ...];
...
in which the diagonals indicated by parenthesis are equal.
Dividing the coefficients of x^(n-1)/(n-1)! in A(x)^(n^3) by n^3, we obtain the following sequence:
[1, 1, 21, 2764, 1249661, 1383968376, 3065126585473, 11913154589356672, 74286423963211939641, 696469981042645688972800, ...].
LOGARITHMIC PROPERTY.
Amazingly, the logarithm of the e.g.f. A(x) is an integer series:
log(A(x)) = x - 3*x^2 - 30*x^3 - 2686*x^4 - 517311*x^5 - 173118807*x^6 - 88535206152*x^7 - 63977172334344*x^8 - 61971659588102940*x^9 - 77470793599569049440*x^10 - 121439997599825393413344*x^11 - 233353875172602479932391040*x^12 - 539638027429765922735002220880*x^13 - 1479049138515818646669055218090480*x^14 - 4742815067612592169849894663392228480*x^15 +...

Links

Crossrefs

Cf. A296173, A296170, A296174, A296176.

Programs

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

Formula

The logarithm of the e.g.f. A(x) is an integer series:
log(A(x)) = Sum{n>=1} A296173(n) * x^n.
E.g.f. A(x) satisfies:
_ 1/n! * d^n/dx^n A(x)^(n^3) = 1/(n-1)! * d^(n-1)/dx^(n-1) A(x)^(n^3) for n>=1, when evaluated at x = 0.
a(n) ~ -sqrt(1-c) * 3^(3*n - 3) * n^(3*n - 3) / (c^n * (3-c)^(2*n - 3) * exp(3*n)), where c = -LambertW(-3*exp(-3)) = -A226750. - Vaclav Kotesovec, Oct 13 2020

A296175 G.f. equals the logarithm of the e.g.f. of A296174.

Original entry on oeis.org

1, -7, -493, -341101, -680813601, -2923660883625, -22996362478599551, -299331006952284448127, -6006951481145880962408552, -176288642409787912257773903552, -7260231964238768891891716773249396, -405879958110794676900559524931590299892, -29968312587171485511980894312242331299164248, -2855987647850204274493781603297327940940773633392
Offset: 1

Views

Author

Paul D. Hanna, Dec 07 2017

Keywords

Comments

E.g.f. G(x) of A296174 satisfies: [x^(n-1)] G(x)^(n^4) = [x^n] G(x)^(n^4) for n>=1.

Examples

			G.f. A(x) = x - 7*x^2 - 493*x^3 - 341101*x^4 - 680813601*x^5 - 2923660883625*x^6 - 22996362478599551*x^7 - 299331006952284448127*x^8 - 6006951481145880962408552*x^9 - 176288642409787912257773903552*x^10 - 7260231964238768891891716773249396*x^11 - 405879958110794676900559524931590299892*x^12 +...
such that
G(x) = exp(A(x)) = 1 + x - 13*x^2/2! - 2999*x^3/3! - 8197751*x^4/4! - 81738176899*x^5/5! - 2105524335759389*x^6/6! - 115916378979693710123*x^7/7! - 12069952631345502122877199*x^8/8! - 2179911119857340269414590758951*x^9/9! - 639738016495616440994202167765715629*x^10/10! +...
satisfies [x^(n-1)] G(x)^(n^4) = [x^n] G(x)^(n^4) for n>=1.
Series_Reversion(A(x)) = x + 7*x^2 + 591*x^3 + 360071*x^4 + 696409901*x^5 + 2958728428011*x^6 + 23164541753169117*x^7 + 300801581861406441263*x^8 +...

Links

Crossrefs

Cf. A296174, A296171, A296173, A296177.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1,n+1, A=concat(A,0); V=Vec(Ser(A)^((#A-1)^4)); A[#A] = (V[#A-1] - V[#A])/(#A-1)^4 ); polcoeff(log(Ser(A)),n)}
for(n=1,30,print1(a(n),", "))

Formula

a(n) ~ -sqrt(1-c) * 2^(8*n - 17/2) * n^(3*n - 9/2) / (sqrt(Pi) * c^n * (4-c)^(3*n - 4) * exp(3*n)), where c = -LambertW(-4*exp(-4)) = 0.07930960512711365643910864... - Vaclav Kotesovec, Oct 13 2020

A296177 G.f. equals the logarithm of the e.g.f. of A296176.

Original entry on oeis.org

1, -15, -6090, -30600650, -593306350650, -31192838317208826, -3652177141294409632400, -836986399841753367052602000, -342157863774785896821739864893375, -232492750600387706453977026534258393375, -248374508240426643818180115122847840121356750, -398845502818641863837604075681689663598753652620750
Offset: 1

Views

Author

Paul D. Hanna, Dec 07 2017

Keywords

Comments

E.g.f. G(x) of A296176 satisfies: [x^(n-1)] G(x)^(n^5) = [x^n] G(x)^(n^5) for n>=1.

Examples

			G.f. A(x) = x - 15*x^2 - 6090*x^3 - 30600650*x^4 - 593306350650*x^5 - 31192838317208826*x^6 - 3652177141294409632400*x^7 - 836986399841753367052602000*x^8 - 342157863774785896821739864893375*x^9 - 232492750600387706453977026534258393375*x^10 +...
such that
G(x) = exp(A(x)) = 1 + x - 29*x^2/2! - 36629*x^3/3! - 734559239*x^4/4! - 71200423546199*x^5/5! - 22459270436075644469*x^6/6! - 18407129959728493123679069*x^7/7! - 33747438879000326056232288023439*x^8/8! - 124162549312926509293620790889452447919*x^9/9! - 843670934957017748849439817665935283173590349*x^10/10! +...
satisfies [x^(n-1)] G(x)^(n^5) = [x^n] G(x)^(n^5) for n>=1.
Series_Reversion(A(x)) = x + 15*x^2 + 6540*x^3 + 31074275*x^4 + 596201157450*x^5 + 31256650109242326*x^6 + 3655957957134009767520*x^7 + 837481638576442353884460435*x^8 +...

Links

Crossrefs

Cf. A296176, A296171, A296173, A296175.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1,n+1, A=concat(A,0); V=Vec(Ser(A)^((#A-1)^5)); A[#A] = (V[#A-1] - V[#A])/(#A-1)^5 ); polcoeff(log(Ser(A)),n)}
for(n=1,30,print1(a(n),", "))

A295813 G.f. A(x) satisfies: G(A(x)) = exp(x), where G(x) equals the e.g.f. of A296172.

Original entry on oeis.org

1, 3, 48, 3271, 575163, 185377116, 93039467356, 66505075585875, 63970743282062646, 79580632411431634441, 124299284968805234137968, 238188439678208173206500760, 549611050835556942751087049225, 1503700734638162443238902233252144, 4814751647416985610768723994195186728, 17841762828286483988438913318683740082187, 75777421917902616009655480827109144353730842
Offset: 1

Views

Author

Paul D. Hanna, Dec 09 2017

Keywords

Comments

E.g.f. G(x) of A296172 satisfies: [x^(n-1)] G(x)^(n^3) = [x^n] G(x)^(n^3) for n>=1.

Examples

			G.f.: A(x) = x + 3*x^2 + 48*x^3 + 3271*x^4 + 575163*x^5 + 185377116*x^6 + 93039467356*x^7 + 66505075585875*x^8 + 63970743282062646*x^9 + 79580632411431634441*x^10 + 124299284968805234137968*x^11 + 238188439678208173206500760*x^12 +...
The series reversion equals the logarithm of the e.g.f. of A296172, which begins:
Series_Reversion(A(x)) = x - 3*x^2 - 30*x^3 - 2686*x^4 - 517311*x^5 - 173118807*x^6 - 88535206152*x^7 - 63977172334344*x^8 - 61971659588102940*x^9 - 77470793599569049440*x^10 - 121439997599825393413344*x^11 - 233353875172602479932391040*x^12 +...+ A296173(n)*x^n +...

Links

Crossrefs

Cf. A296172, A296173, A295812, A295814.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1, n+1, A=concat(A, 0); V=Vec(Ser(A)^((#A-1)^3)); A[#A] = (V[#A-1] - V[#A])/(#A-1)^3 ); polcoeff(serreverse(log(Ser(A))), n)}
for(n=1, 30, print1(a(n), ", "))

Formula

G.f. is the series reversion of the logarithm of the e.g.f. of A296172.
a(n) ~ sqrt(1-c) * 3^(3*n - 3) * n^(2*n - 7/2) / (sqrt(2*Pi) * c^n * (3-c)^(2*n - 3) * exp(2*n)), where c = -LambertW(-3*exp(-3)) = -A226750. - Vaclav Kotesovec, Oct 13 2020
Showing 1-5 of 5 results.

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