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.

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!.

A155804 E.g.f. satisfies: A(x) = Sum_{n>=0} x^n/n! * A(x)^(n(n-1)/2).

Original entry on oeis.org

1, 1, 1, 4, 19, 161, 1606, 21022, 323485, 5874913, 122077756, 2871573596, 75437801539, 2193468714373, 70020045331510, 2437979768144026, 92073099488632441, 3753886179551636513, 164556499026975482008
Offset: 0

Views

Author

Paul D. Hanna, Jan 27 2009

Keywords

Examples

			E.g.f.: A(x) = 1 + x + x^2/2! + 4*x^3/3! + 19*x^4/4! + 161*x^5/5! +...
where e.g.f. A(x) satisfies:
A(x) = 1 + x + x^2/2!*A(x) + x^3/3!*A(x)^3 + x^4/4!*A(x)^6 + x^5/5!*A(x)^10 +...
Let B(x) = A(x*B(x)) be the e.g.f. of A155805 then:
B(x) = 1 + x*B(x) + x^2/2!*B(x)^3 + x^3/3!*B(x)^6 + x^4/4!*B(x)^10 +...
B(x) = 1 + x + 3*x^2/2! + 19*x^3/3! + 191*x^4/4! + 2656*x^5/5! + 47392*x^6/6! +...

Crossrefs

Cf. A155805, A155806, A155807, A107590, A219358.

Programs

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

Formula

E.g.f. satisfies: A(x) = B(x/A(x)) and A(x*B(x)) = B(x) where B(x) satisfies:
B(x) = Sum_{n>=0} x^n/n! * B(x)^(n*(n+1)/2) and is the e.g.f. of A155805.

A155805 E.g.f. satisfies: A(x) = Sum_{n>=0} x^n/n! * A(x)^(n(n+1)/2).

Original entry on oeis.org

1, 1, 3, 19, 191, 2656, 47392, 1034335, 26721781, 798018616, 27058991246, 1027237384009, 43172232488959, 1990253576425960, 99871804451808040, 5419775866582473211, 316301430225674131433, 19756213549154356027408
Offset: 0

Views

Author

Paul D. Hanna, Jan 27 2009

Keywords

Examples

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 19*x^3/3! + 191*x^4/4! + 2656*x^5/5! +...
where e.g.f. A(x) satisfies:
A(x) = 1 + x*A(x) + x^2/2!*A(x)^3 + x^3/3!*A(x)^6 + x^4/4!*A(x)^10 +...
Let B(x) = A(x/B(x)) be the e.g.f. of A155804 then:
B(x) = 1 + x + x^2/2!*B(x) + x^3/3!*B(x)^3 + x^4/4!*B(x)^6 + x^5/5!*B(x)^10 +...
B(x) = 1 + x + x^2/2! + 4*x^3/3! + 19*x^4/4! + 161*x^5/5! + 1606*x^6/6! +...

Crossrefs

Cf. A155804, A155806, A155807, A107591, A219359.

Programs

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

Formula

E.g.f. satisfies: A(x) = B(x*A(x)) and A(x/B(x)) = B(x) where B(x) satisfies:
B(x) = Sum_{n>=0} x^n/n! * B(x)^(n*(n-1)/2) and is the e.g.f. of A155804.

A155807 E.g.f. satisfies: A(x) = Sum_{n>=0} x^n/n! * A(x)^(n(n+1)).

Original entry on oeis.org

1, 1, 5, 55, 969, 23661, 741013, 28363707, 1284098609, 67149601273, 3984121444581, 264485848799679, 19426332734137849, 1564277403496216293, 137040382838351173301, 12977244383702330201731
Offset: 0

Views

Author

Paul D. Hanna, Jan 27 2009

Keywords

Examples

			E.g.f.: A(x) = 1 + x + 5*x^2/2! + 55*x^3/3! + 969*x^4/4! + 23661*x^5/5! +...
where e.g.f. A(x) satisfies:
A(x) = 1 + x*A(x)^2 + x^2/2!*A(x)^6 + x^3/3!*A(x)^12 + x^4/4!*A(x)^20 +...
Let B(x) = A(x/B(x)) be the e.g.f. of A155806 then:
B(x) = 1 + x*B(x) + x^2/2!*B(x)^4 + x^3/3!*B(x)^9 + x^4/4!*B(x)^16 +...
B(x) = 1 + x + 3*x^2/2! + 22*x^3/3! + 269*x^4/4! + 4616*x^5/5! + 102847*x^6/6! +...

Crossrefs

Cf. A155804, A155805, A155806.

Programs

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

Formula

E.g.f. satisfies: A(x) = B(x*A(x)) and A(x/B(x)) = B(x) where B(x) satisfies:
B(x) = Sum_{n>=0} x^n/n! * B(x)^(n^2) and is the e.g.f. of A155806.

A192036 E.g.f. satisfies: A(x) = Sum_{n>=0} x^n*A(n*x)^n/n!.

Original entry on oeis.org

1, 1, 3, 22, 317, 7976, 329167, 21511036, 2187830521, 343670351392, 83118756921371, 30891910810811084, 17606061819337679173, 15347380239670729742272, 20404520526924833144453623, 41254672227383167503175726876, 126484184787351358506375259745393
Offset: 0

Views

Author

Paul D. Hanna, Jun 21 2011

Keywords

Examples

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 22*x^3/3! + 317*x^4/4! + 7976*x^5/5! +...
The e.g.f. satisfies:
A(x) = 1 + x*A(x) + x^2*A(2*x)^2/2! + x^3*A(3*x)^3/3! + x^4*A(4*x)^4/4! +...

Crossrefs

Cf. A125281, A155806, A218682.

Programs

  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=1+sum(m=1,n,x^m*subst(A,x,m*x+x*O(x^(n)))^m/m!));n!*polcoeff(A,n)}
for(n=0,20,print1(a(n),", "))
Showing 1-5 of 5 results.

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