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.

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A361544 a(n) = A361540(n,1) for n >= 1, a column of triangle A361540.

Original entry on oeis.org

1, 4, 39, 604, 12625, 332766, 10574725, 393171416, 16744363569, 803841993370, 42957812253301, 2529951235854516, 162852898603253209, 11378885054925777494, 858009440175419213445, 69471138931959493061296, 6013997809048628612191585, 554545575488282609142617778
Offset: 1

Views

Author

Paul D. Hanna, Mar 20 2023

Keywords

Comments

E.g.f. F(x,y) of triangle A361540 satisfies the following.
(1) F(x,y) = Sum_{n>=0} (F(x,y)^n + y)^n * x^n/n!.
(2) F(x,y) = Sum_{n>=0} F(x,y)^(n^2) * exp(y*x*F(x,y)^n) * x^n/n!.
The column next to this one in triangle A361540 has e.g.f. G(x) = Sum_{n>=0} G(x)^(n^2)*x^n/n!.

Examples

			E.g.f.: A(x) = x + 4*x^2/2! + 39*x^3/3! + 604*x^4/4! + 12625*x^5/5! + 332766*x^6/6! + 10574725*x^7/7! + 393171416*x^8/8! + 16744363569*x^9/9! + 803841993370*x^10/10! + ... + a(n)*x^n/n! + ...
a(n) is divisible by n, where a(n)/n begins
[1, 2, 13, 151, 2525, 55461, 1510675, 49146427, 1860484841, ...].

Links

Crossrefs

Cf. A361540, A185298.

Programs

  • PARI
    /* E.g.f. of triangle A361540 is F(x,y) = Sum_{n>=0} (F(x,y)^n + y)^n * x^n/n! */
{A361540(n,k) = my(F = 1); for(i=1,n, F = sum(m=0, n, (F^m + y +x*O(x^n))^m * x^m/m! )); n!*polcoeff(polcoeff(F, n,x),k,y)}
for(n=1, 20, print1(A361540(n,1), ", "))

A361539 a(n) = A361540(n, n-2) for n >= 2, a diagonal of triangle A361540.

Original entry on oeis.org

3, 39, 426, 4550, 50085, 577731, 7022596, 90148860, 1222753815, 17515226465, 264663151038, 4212100028994, 70475063838361, 1237144088015535, 22735980569119560, 436467520716475064, 8733235757816095083, 181740089309026259565, 3925458146197916823970
Offset: 2

Views

Author

Paul D. Hanna, Mar 20 2023

Keywords

Comments

E.g.f. F(x,y) of triangle A361540 satisfies the following.
(1) F(x,y) = Sum_{n>=0} (F(x,y)^n + y)^n * x^n/n!.
(2) F(x,y) = Sum_{n>=0} F(x,y)^(n^2) * exp(y*x*F(x,y)^n) * x^n/n!.
The diagonal above this one in triangle A361540 has e.g.f. x*exp(x)*exp(x*exp(x)).

Examples

			E.g.f. A(x) = 3*x^2/2! + 39*x^3/3! + 426*x^4/4! + 4550*x^5/5! + 50085*x^6/6! + 577731*x^7/7! + 7022596*x^8/8! + 90148860*x^9/9! + 1222753815*x^10/10! + ...

Links

Crossrefs

Cf. A361540.

Programs

  • PARI
    /* E.g.f. of triangle A361540 is F(x,y) = Sum_{n>=0} (F(x,y)^n + y)^n * x^n/n! */
{A361540(n,k) = my(A = 1); for(i=1,n, A = sum(m=0, n, (A^m + y +x*O(x^n))^m * x^m/m! )); n!*polcoeff(polcoeff(A, n,x),k,y)}
for(n=2, 20, print1(A361540(n,n-2), ", "))

A361549 a(n) = A361540(n,2) for n >= 2, a column of triangle A361540.

Original entry on oeis.org

1, 18, 426, 12040, 401355, 15456756, 676130644, 33151425840, 1802216703285, 107652497473180, 7012494336544686, 494963689847333928, 37648456802884402111, 3071415347513049808740, 267644521958509484952360, 24822151072519637091258976, 2442314922307988498911793385
Offset: 2

Views

Author

Paul D. Hanna, Mar 20 2023

Keywords

Comments

E.g.f. F(x,y) of triangle A361540 satisfies the following.
(1) F(x,y) = Sum_{n>=0} (F(x,y)^n + y)^n * x^n/n!.
(2) F(x,y) = Sum_{n>=0} F(x,y)^(n^2) * exp(y*x*F(x,y)^n) * x^n/n!.
Column 0 near to this one in triangle A361540 has e.g.f. G(x) = Sum_{n>=0} G(x)^(n^2)*x^n/n!.

Examples

			E.g.f.: A(x) = x^2/2! + 18*x^3/3! + 426*x^4/4! + 12040*x^5/5! + 401355*x^6/6! + 15456756*x^7/7! + 676130644*x^8/8! + 33151425840*x^9/9! + 1802216703285*x^10/10! + ... + a(n)*x^n/n! + ...
a(n) is divisible by n*(n-1)/2, where a(n)*2/(n*(n-1)) begins
[1, 6, 71, 1204, 26757, 736036, 24147523, 920872940, 40049260073, ...].

Links

Crossrefs

Cf. A361540, A361544.

Programs

  • PARI
    /* E.g.f. of triangle A361540 is F(x,y) = Sum_{n>=0} (F(x,y)^n + y)^n * x^n/n! */
{A361540(n,k) = my(F = 1); for(i=1,n, F = sum(m=0, n, (F^m + y +x*O(x^n))^m * x^m/m! )); n!*polcoeff(polcoeff(F, n,x),k,y)}
for(n=2, 20, print1(A361540(n,2), ", "))

A361688 a(n) = A361540(2*n,n) / binomial(2*n,n) for n >= 0.

Original entry on oeis.org

1, 2, 71, 10915, 4063645, 2842101221, 3255178907803, 5605980824208871, 13710496284516264953, 45746570903514799640905, 202291094041887013214628871, 1160411497892246920315488823067, 8496377826955803443098054623140629, 78398366060939693412478828210386035725
Offset: 0

Views

Author

Paul D. Hanna, Mar 20 2023

Keywords

Comments

E.g.f. F(x,y) of triangle A361540 satisfies the following.
(1) F(x,y) = Sum_{n>=0} (F(x,y)^n + y)^n * x^n/n!.
(2) F(x,y) = Sum_{n>=0} F(x,y)^(n^2) * exp(y*x*F(x,y)^n) * x^n/n!.
This sequence equals the central terms of triangle A361540 divided by the central binomial coefficients A000984.

Examples

			E.g.f. A(x) = 1 + 2*x + 71*x^2/2! + 10915*x^3/3! + 4063645*x^4/4! + 2842101221*x^5/5! + 3255178907803*x^6/6! + 5605980824208871*x^7/7! + 13710496284516264953*x^8/8! + ... + a(n)*x^n/n! + ...

Links

Crossrefs

Cf. A361540, A000984.

Programs

  • PARI
    /* E.g.f. of triangle A361540 is F(x,y) = Sum_{n>=0} (F(x,y)^n + y)^n * x^n/n! */
{A361540(n,k) = my(F = 1); for(i=1,n, F = sum(m=0, n, (F^m + y +x*O(x^n))^m * x^m/m! )); n!*polcoeff(polcoeff(F, n,x),k,y)}
{a(n) = A361540(2*n,n)/binomial(2*n,n)}
for(n=0, 15, print1(a(n), ", "))

A361053 Expansion of e.g.f. A(x) satisfying A(x) = Sum_{n>=0} (A(x)^n + 2)^n * x^n/n!.

Original entry on oeis.org

1, 3, 15, 180, 3933, 122778, 5024727, 255694050, 15594132825, 1110807585090, 90665847445059, 8355178654847874, 859198582766876661, 97668423691415577666, 12177783763614287432847, 1654751006054203510476882, 243720706148230009547388465, 38730619011753683906970442626
Offset: 0

Views

Author

Paul D. Hanna, Feb 28 2023

Keywords

Examples

			E.g.f.: A(x) = 1 + 3*x + 15*x^2/2! + 180*x^3/3! + 3933*x^4/4! + 122778*x^5/5! + 5024727*x^6/6! + 255694050*x^7/7! + 15594132825*x^8/8! +...
where the e.g.f. satisfies the following series identity:
A(x) = 1 + (A(x) + 2)*x + (A(x)^2 + 2)^2*x^2/2! + (A(x)^3 + 2)^3*x^3/3! + (A(x)^4 + 2)^4*x^4/4! + ... + (A(x)^n + 2)^n * x^n/n! + ...
and
A(x) = exp(2*x) + A(x)*exp(2*x*A(x))*x + A(x)^4*exp(2*x*A(x)^2)*x^2/2! + A(x)^9*exp(2*x*A(x)^3)*x^3/3! + A(x)^16*exp(2*x*A(x)^4)*x^4/4! + ... + A(x)^(n^2) * exp(2*x*A(x)^n) * x^n/n! + ...

Links

Crossrefs

Cf. A202999, A361054, A361055, A361056, A361057, A203013.
Cf. A361540.

Programs

  • PARI
    /* E.g.f.: Sum_{n>=0} (A(x)^n + 2)^n * x^n/n! */
{a(n) = my(A = 1); for(i=1,n, A = sum(m=0, n, (A^m + 2 +x*O(x^n))^m * x^m/m! )); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
  • PARI
    /* E.g.f.: Sum_{n>=0} A(x)^(n^2) * exp(2*x*A(x)^n) * x^n/n! */
{a(n) = my(A=1); for(i=1,n, A = sum(m=0, n, (A +x*O(x^n))^(m^2) * exp(2*x*A^m +x*O(x^n)) * x^m/m! )); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))

Formula

E.g.f. A(x) = Sum_{n>=0} a(n) * x^n/n! may be defined as follows.
(1) A(x) = Sum_{n>=0} (A(x)^n + 2)^n * x^n/n!.
(2) A(x) = Sum_{n>=0} A(x)^(n^2) * exp(2*x*A(x)^n) * x^n/n!.
a(n) = 0 (mod 3) for n > 0.
a(n) = Sum_{k=0..n} A361540(n,k) * 2^k. - Paul D. Hanna, Mar 20 2023

A361054 Expansion of e.g.f. A(x) satisfying A(x) = Sum_{n>=0} (A(x)^n + 3)^n * x^n/n!.

Original entry on oeis.org

1, 4, 24, 328, 8480, 316064, 15448000, 940586560, 68773511680, 5883198833152, 577566163260416, 64112172571384832, 7953180924959641600, 1092205827724943429632, 164769061745517773774848, 27131359440809990936141824, 4850231804845681441360707584, 937096082325039305880612503552
Offset: 0

Views

Author

Paul D. Hanna, Feb 28 2023

Keywords

Examples

			E.g.f.: A(x) = 1 + 4*x + 24*x^2/2! + 328*x^3/3! + 8480*x^4/4! + 316064*x^5/5! + 15448000*x^6/6! + 940586560*x^7/7! + 68773511680*x^8/8! +...
where the e.g.f. satisfies the following series identity:
A(x) = 1 + (A(x) + 3)*x + (A(x)^2 + 3)^2*x^2/2! + (A(x)^3 + 3)^3*x^3/3! + (A(x)^4 + 3)^4*x^4/4! + ... + (A(x)^n + 3)^n * x^n/n! + ...
and
A(x) = exp(3*x) + A(x)*exp(3*x*A(x))*x + A(x)^4*exp(3*x*A(x)^2)*x^2/2! + A(x)^9*exp(3*x*A(x)^3)*x^3/3! + A(x)^16*exp(3*x*A(x)^4)*x^4/4! + ... + A(x)^(n^2) * exp(3*x*A(x)^n) * x^n/n! + ...

Links

Crossrefs

Cf. A202999, A361053, A361055, A361056, A361057, A203013.
Cf. A361540.

Programs

  • PARI
    /* E.g.f.: Sum_{n>=0} (A(x)^n + 3)^n * x^n/n! */
{a(n) = my(A = 1); for(i=1,n, A = sum(m=0, n, (A^m + 3 +x*O(x^n))^m * x^m/m! )); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
  • PARI
    /* E.g.f.: Sum_{n>=0} A(x)^(n^2) * exp(3*x*A(x)^n) * x^n/n! */
{a(n) = my(A=1); for(i=1,n, A = sum(m=0, n, (A +x*O(x^n))^(m^2) * exp(3*x*A^m +x*O(x^n)) * x^m/m! )); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))

Formula

E.g.f. A(x) = Sum_{n>=0} a(n) * x^n/n! may be defined as follows.
(1) A(x) = Sum_{n>=0} (A(x)^n + 3)^n * x^n/n!.
(2) A(x) = Sum_{n>=0} A(x)^(n^2) * exp(3*x*A(x)^n) * x^n/n!.
a(n) = 0 (mod 4) for n > 0.
a(n) = Sum_{k=0..n} A361540(n,k) * 3^k. - Paul D. Hanna, Mar 20 2023

A361055 Expansion of e.g.f. A(x) satisfying A(x) = Sum_{n>=0} (A(x)^n + 4)^n * x^n/n!.

Original entry on oeis.org

1, 5, 35, 530, 15645, 673100, 37951975, 2668045700, 225591547225, 22347122264900, 2543582111665875, 327736278022956500, 47245927138947731125, 7548695252947520166500, 1326483608786914301185375, 254733442821907695977652500, 53175506363950820566794680625, 12012490474019349963485905242500
Offset: 0

Views

Author

Paul D. Hanna, Feb 28 2023

Keywords

Examples

			E.g.f.: A(x) = 1 + 5*x + 35*x^2/2! + 530*x^3/3! + 15645*x^4/4! + 673100*x^5/5! + 37951975*x^6/6! + 2668045700*x^7/7! + 225591547225*x^8/8! +...
where the e.g.f. satisfies the following series identity:
A(x) = 1 + (A(x) + 4)*x + (A(x)^2 + 4)^2*x^2/2! + (A(x)^3 + 4)^3*x^3/3! + (A(x)^4 + 4)^4*x^4/4! + ... + (A(x)^n + 4)^n * x^n/n! + ...
and
A(x) = exp(4*x) + A(x)*exp(4*x*A(x))*x + A(x)^4*exp(4*x*A(x)^2)*x^2/2! + A(x)^9*exp(4*x*A(x)^3)*x^3/3! + A(x)^16*exp(4*x*A(x)^4)*x^4/4! + ... + A(x)^(n^2) * exp(4*x*A(x)^n) * x^n/n! + ...

Links

Crossrefs

Cf. A202999, A361053, A361054, A361056, A361057, A203013.
Cf. A361540.

Programs

  • PARI
    /* E.g.f.: Sum_{n>=0} (A(x)^n + 4)^n * x^n/n! */
{a(n) = my(A = 1); for(i=1,n, A = sum(m=0, n, (A^m + 4 +x*O(x^n))^m * x^m/m! )); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
  • PARI
    /* E.g.f.: Sum_{n>=0} A(x)^(n^2) * exp(4*x*A(x)^n) * x^n/n! */
{a(n) = my(A=1); for(i=1,n, A = sum(m=0, n, (A +x*O(x^n))^(m^2) * exp(4*x*A^m +x*O(x^n)) * x^m/m! )); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))

Formula

E.g.f. A(x) = Sum_{n>=0} a(n) * x^n/n! may be defined as follows.
(1) A(x) = Sum_{n>=0} (A(x)^n + 4)^n * x^n/n!.
(2) A(x) = Sum_{n>=0} A(x)^(n^2) * exp(4*x*A(x)^n) * x^n/n!.
a(n) = 0 (mod 5) for n > 0.
a(n) = Sum_{k=0..n} A361540(n,k) * 4^k. - Paul D. Hanna, Mar 20 2023

A361057 Expansion of e.g.f. A(x) satisfying A(x) = Sum_{n>=0} (3*A(x)^n + 1)^n * x^n/n!.

Original entry on oeis.org

1, 4, 40, 1000, 42208, 2511904, 194701888, 18644964160, 2128895802880, 282664859507200, 42830926407126016, 7299282818219035648, 1382930912338770866176, 288548709643121903915008, 65787364162207649519116288, 16282501210870115738111156224, 4350458941547832791800523653120
Offset: 0

Views

Author

Paul D. Hanna, Feb 28 2023

Keywords

Examples

			E.g.f.: A(x) = 1 + 4*x + 40*x^2/2! + 1000*x^3/3! + 42208*x^4/4! + 2511904*x^5/5! + 194701888*x^6/6! + 18644964160*x^7/7! + 2128895802880*x^8/8! +...
where the e.g.f. satisfies the following series identity:
A(x) = 1 + (3*A(x) + 1)*x + (3*A(x)^2 + 1)^2*x^2/2! + (3*A(x)^3 + 1)^3*x^3/3! + (3*A(x)^4 + 1)^4*x^4/4! + ... + (3*A(x)^n + 1)^n * x^n/n! + ...
and
A(x) = exp(x) + A(x)*exp(x*A(x))*3*x + A(x)^4*exp(x*A(x)^2)*3^2*x^2/2! + A(x)^9*exp(x*A(x)^3)*3^3*x^3/3! + A(x)^16*exp(x*A(x)^4)*3^4*x^4/4! + ... + A(x)^(n^2) * exp(x*A(x)^n) * 3^n * x^n/n! + ...

Links

Crossrefs

Cf. A202999, A361053, A361054, A361055, A361056, A203013.
Cf. A361540.

Programs

  • PARI
    /* E.g.f.: Sum_{n>=0} (3*A(x)^n + 1)^n * x^n/n! */
{a(n) = my(A = 1); for(i=1,n, A = sum(m=0, n, (3*A^m + 1 +x*O(x^n))^m * x^m/m! )); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
  • PARI
    /* E.g.f.: Sum_{n>=0} A(x)^(n^2) * exp(x*A(x)^n) * 3^n * x^n/n! */
{a(n) = my(A=1); for(i=1,n, A = sum(m=0, n, (A +x*O(x^n))^(m^2) * exp(x*A^m +x*O(x^n)) * 3^m * x^m/m! )); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))

Formula

E.g.f. A(x) = Sum_{n>=0} a(n) * x^n/n! may be defined as follows.
(1) A(x) = Sum_{n>=0} (3*A(x)^n + 1)^n * x^n/n!.
(2) A(x) = Sum_{n>=0} A(x)^(n^2) * exp(x*A(x)^n) * 3^n * x^n/n!.
a(n) = 0 (mod 4) for n > 0.
a(n) = Sum_{k=0..n} A361540(n,k) * 3^(n-k). - Paul D. Hanna, Mar 20 2023

A202999 Expansion of e.g.f. A(x) satisfying A(x) = Sum_{n>=0} (A(x)^n + 1)^n * x^n/n!.

Original entry on oeis.org

1, 2, 8, 80, 1392, 34352, 1108576, 44340704, 2119928320, 118111781888, 7524579815424, 540141484897280, 43182173208678400, 3808622859938226176, 367715812648914460672, 38610662734158029938688, 4384921058923036753723392, 536091721631513000647393280
Offset: 0

Views

Author

Paul D. Hanna, Dec 27 2011

Keywords

Examples

			E.g.f.: A(x) = 1 + 2*x + 8*x^2/2! + 80*x^3/3! + 1392*x^4/4! + 34352*x^5/5! +...
where the e.g.f. satisfies following series identity:
A(x) = 1 + (A(x)+1)*x + (A(x)^2+1)^2*x^2/2! + (A(x)^3+1)^3*x^3/3! + (A(x)^4+1)^4*x^4/4! +...
A(x) = exp(x) + A(x)*exp(x*A(x))*x + A(x)^4*exp(x*A(x)^2)*x^2/2! + A(x)^9*exp(x*A(x)^3)*x^3/3! + A(x)^16*exp(x*A(x)^4)*x^4/4! +...

Crossrefs

Cf. A361053, A361054, A361055, A361057, A203013.
Cf. A361540.

Programs

  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=sum(k=0, n, (A^k+1+x*O(x^n))^k*x^k/k!));n!*polcoeff(A, n)}
  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=sum(k=0, n, A^(k^2)*exp(A^k*x+x*O(x^n))*x^k/k!));n!*polcoeff(A, n)}

Formula

E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies the following.
(1) A(x) = Sum_{n>=0} (A(x)^n + 1)^n * x^n/n!.
(2) A(x) = Sum_{n>=0} A(x)^(n^2) * exp(x*A(x)^n) * x^n/n!.
a(n) = Sum_{k=0..n} A361540(n,k). - Paul D. Hanna, Mar 20 2023

A203013 Expansion of e.g.f. A(x) satisfying A(x) = Sum_{n>=0} (2*A(x)^n - 1)^n * x^n/n!.

Original entry on oeis.org

1, 1, 5, 55, 993, 24871, 802873, 31793035, 1493163745, 81186783535, 5018214016041, 347636382949747, 26685235607680081, 2248760378885064487, 206430769607981879353, 20507793044444903462251, 2192507508237447321800385, 251034864831917236610746207
Offset: 0

Views

Author

Paul D. Hanna, Dec 27 2011

Keywords

Examples

			E.g.f.: A(x) = 1 + x + 5*x^2/2! + 55*x^3/3! + 993*x^4/4! + 24871*x^5/5! +...
where the e.g.f. satisfies following series identity:
A(x) = 1 + (2*A(x)-1)*x + (2*A(x)^2-1)^2*x^2/2! + (2*A(x)^3-1)^3*x^3/3! + (2*A(x)^4-1)^4*x^4/4! +...
is equal to
A(x) = exp(-x) + 2*A(x)*exp(-x*A(x))*x + 2^2*A(x)^4*exp(-x*A(x)^2)*x^2/2! + 2^3*A(x)^9*exp(-x*A(x)^3)*x^3/3! + 2^4*A(x)^16*exp(-x*A(x)^4)*x^4/4! +...

Crossrefs

Cf. A202999, A361053, A361054, A361055, A361056, A361057.
Cf. A361540.

Programs

  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=sum(k=0, n, (2*A^k-1+x*O(x^n))^k*x^k/k!));n!*polcoeff(A, n)}
  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=sum(k=0, n, 2^k*A^(k^2)*exp(-A^k*x+x*O(x^n))*x^k/k!));n!*polcoeff(A, n)}

Formula

E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies the following.
(1) A(x) = Sum_{n>=0} (2*A(x)^n - 1)^n * x^n/n!.
(2) A(x) = Sum_{n>=0} 2^n * A(x)^(n^2) * exp(-x*A(x)^n) * x^n/n!.
a(n) = Sum_{k=0..n} A361540(n,k) * 2^(n-k) * (-1)^k. - Paul D. Hanna, Mar 20 2023
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