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 1-7 of 7 results.

A361540 Expansion of e.g.f. A(x,y) satisfying A(x,y) = Sum_{n>=0} (A(x,y)^n + y)^n * x^n/n!, as a triangle read by rows.

Original entry on oeis.org

1, 1, 1, 3, 4, 1, 22, 39, 18, 1, 269, 604, 426, 92, 1, 4616, 12625, 12040, 4550, 520, 1, 102847, 332766, 401355, 218300, 50085, 3222, 1, 2824816, 10574725, 15456756, 11017895, 3867080, 577731, 21700, 1, 92355769, 393171416, 676130644, 597596216, 284455150, 69038984, 7022596, 157544, 1
Offset: 0

Views

Author

Paul D. Hanna, Mar 20 2023

Keywords

Comments

A202999(n) = Sum_{k=0..n} T(n,k).
A361053(n) = Sum_{k=0..n} T(n,k) * 2^k.
A361054(n) = Sum_{k=0..n} T(n,k) * 3^k.
A361055(n) = Sum_{k=0..n} T(n,k) * 4^k.
A361056(n) = Sum_{k=0..n} T(n,k) * 2^(n-k).
A361057(n) = Sum_{k=0..n} T(n,k) * 3^(n-k).
A203013(n) = Sum_{k=0..n} T(n,k) * 2^(n-k) * (-1)^k.
A155806(n) = T(n,0) for n >= 0; e.g.f. G(x) = Sum_{n>=0} G(x)^(n^2)*x^n/n!.
A361544(n) = T(n,1) for n >= 1.
A361549(n) = T(n,2) for n >= 2.
A185298(n) = T(n,n-1) for n >= 1; e.g.f. x*exp(x)*exp(x*exp(x)).
A361539(n) = T(n,n-2) for n >= 2.
A361688(n) = T(2*n,n) / binomial(2*n,n) for n >= 0.

Examples

			E.g.f. A(x,y) = 1 + (y + 1)*x + (y^2 + 4*y + 3)*x^2/2! + (y^3 + 18*y^2 + 39*y + 22)*x^3/3! + (y^4 + 92*y^3 + 426*y^2 + 604*y + 269)*x^4/4! + (y^5 + 520*y^4 + 4550*y^3 + 12040*y^2 + 12625*y + 4616)*x^5/5! + (y^6 + 3222*y^5 + 50085*y^4 + 218300*y^3 + 401355*y^2 + 332766*y + 102847)*x^6/6! + (y^7 + 21700*y^6 + 577731*y^5 + 3867080*y^4 + 11017895*y^3 + 15456756*y^2 + 10574725*y + 2824816)*x^7/7! + (y^8 + 157544*y^7 + 7022596*y^6 + 69038984*y^5 + 284455150*y^4 + 597596216*y^3 + 676130644*y^2 + 393171416*y + 92355769)*x^8/8! + ...
This triangle of coefficients T(n,k) of x^n*y^k in e.g.f. A(x,y) begins:
[1];
[1, 1];
[3, 4, 1];
[22, 39, 18, 1];
[269, 604, 426, 92, 1];
[4616, 12625, 12040, 4550, 520, 1];
[102847, 332766, 401355, 218300, 50085, 3222, 1];
[2824816, 10574725, 15456756, 11017895, 3867080, 577731, 21700, 1];
[92355769, 393171416, 676130644, 597596216, 284455150, 69038984, 7022596, 157544, 1];
[3506278528, 16744363569, 33151425840, 35028273756, 21134516256, 7193104758, 1262445744, 90148860, 1224576, 1]; ...
RELATED TABLE.
The elements of this triangle T(n,k) when divided by binomial(n,k) yields the related triangle:
[1];
[1, 1];
[3, 2, 1];
[22, 13, 6, 1];
[269, 151, 71, 23, 1];
[4616, 2525, 1204, 455, 104, 1];
[102847, 55461, 26757, 10915, 3339, 537, 1];
[2824816, 1510675, 736036, 314797, 110488, 27511, 3100, 1];
[92355769, 49146427, 24147523, 10671361, 4063645, 1232839, 250807, 19693, 1];
[3506278528, 1860484841, 920872940, 417003259, 167734256, 57088133, 15029116, 2504135, 136064, 1]; ...

Links

Crossrefs

Cf. A202999 (y=1), A361053 (y=2), A361054 (y=3), A361055 (y=4), A361056, A361057, A203013.
Cf. A155806 (T(n,0)), A361544 (T(n,1)), A361549 (T(n,2)), A185298 (T(n,n-1)), A361539 (T(n,n-2)), A361688 (T(2*n,n)/C(2*n,n)).

Programs

  • PARI
    /* E.g.f. A(x,y) = Sum_{n>=0} (A(x,y)^n + y)^n * x^n/n! */
{T(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=0, 12, for(k=0, n, print1(T(n,k), ", ")); print(" "))
  • PARI
    /* E.g.f. A(x,y) = Sum_{n>=0} A(x,y)^(n^2) * exp(y*x*A(x,y)^n) * x^n/n! */
{T(n,k) = my(A=1); for(i=1,n, A = sum(m=0, n, (A +x*O(x^n))^(m^2) * exp(y*x*A^m +x*O(x^n)) * x^m/m! )); n!*polcoeff(polcoeff(A, n,x),k,y)}
for(n=0, 12, for(k=0, n, print1(T(n,k), ", ")); print(" "))

Formula

E.g.f. A(x,y) = Sum_{n>=0} Sum_{k=0..n} T(n,k)*x^n*y^k/n! may be defined as follows.
(1) A(x,y) = Sum_{n>=0} (A(x,y)^n + y)^n * x^n/n!.
(2) A(x,y) = Sum_{n>=0} A(x,y)^(n^2) * exp(y*x*A(x,y)^n) * x^n/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

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

A361056 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, 3, 21, 369, 11025, 465273, 25605585, 1742552325, 141496457985, 13368820514769, 1442273097241809, 175090338669687741, 23642282811004895745, 3517444221383606541849, 572114802197326599160497, 101067684833728895205914757, 19284211878473628720362002689
Offset: 0

Views

Author

Paul D. Hanna, Feb 28 2023

Keywords

Examples

			E.g.f.: A(x) = 1 + 3*x + 21*x^2/2! + 369*x^3/3! + 11025*x^4/4! + 465273*x^5/5! + 25605585*x^6/6! + 1742552325*x^7/7! + 141496457985*x^8/8! +...
where the e.g.f. satisfies the 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! + ... + (2*A(x)^n + 1)^n * x^n/n! + ...
and
A(x) = exp(x) + A(x)*exp(x*A(x))*2*x + A(x)^4*exp(x*A(x)^2)*2^2*x^2/2! + A(x)^9*exp(x*A(x)^3)*2^3*x^3/3! + A(x)^16*exp(x*A(x)^4)*2^4*x^4/4! + ... + A(x)^(n^2) * exp(x*A(x)^n) * 2^n * x^n/n! + ...

Links

Crossrefs

Cf. A202999, A361053, A361054, A361055, A361057, A203013.
Cf. A361540.

Programs

  • PARI
    /* E.g.f.: Sum_{n>=0} (2*A(x)^n + 1)^n * x^n/n! */
{a(n) = my(A = 1); for(i=1,n, A = sum(m=0, n, (2*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) * 2^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)) * 2^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} (2*A(x)^n + 1)^n * x^n/n!.
(2) A(x) = Sum_{n>=0} A(x)^(n^2) * exp(x*A(x)^n) * 2^n * x^n/n!.
a(n) = 0 (mod 3) for n > 0.
a(n) = Sum_{k=0..n} A361540(n,k) * 2^(n-k). - Paul D. Hanna, Mar 20 2023
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.