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 1-6 of 6 results.

A180719 Logarithmic derivative of A180718.

Original entry on oeis.org

1, 5, 16, 61, 226, 884, 3543, 14429, 59623, 248950, 1049159, 4454356, 19032976, 81769735, 352967821, 1529948477, 6655903632, 29050257899, 127162016206, 558088733406, 2455157735151, 10824115727199, 47814658900427
Offset: 1

Views

Author

Paul D. Hanna, Sep 24 2010

Keywords

Examples

			L.g.f.: L(x) = x + 5*x^2/2 + 16*x^3/3 + 61*x^4/4 + 226*x^5/5 +...
which equals the sum of the series:
L(x) = (1 + x)^2*x
+ (1 + 4*x + x^2)^2*x^2/2
+ (1 + 9*x + 9*x^2 + x^3)^2*x^3/3
+ (1 + 16*x + 36*x^2 + 16*x^3 + x^4)^2*x^4/4
+ (1 + 25*x + 100*x^2 + 100*x^3 + 25*x^4 + x^5)^2*x^5/5
+ (1 + 36*x + 225*x^2 + 400*x^3 + 225*x^4 + 36*x^5 + x^6)^2*x^6/6 +...
where exponentiation yields the integer series:
exp(L(x)) = 1 + x + 3*x^2 + 8*x^3 + 25*x^4 + 80*x^5 + 271*x^6 + 952*x^7 + 3443*x^8 + 12758*x^9 + 48212*x^10 +...+ A180718(n)*x^n/n +...

Crossrefs

Cf. A180718.

Programs

  • PARI
    {a(n)=n*polcoeff(sum(m=1,n,sum(k=0,m,binomial(m,k)^2*x^k)^2*x^m/m)+x*O(x^n),n)}

Formula

L.g.f.: L(x) = Sum_{n>=0} [ Sum_{k=0..n} C(n,k)^2*x^k ]^2*x^n/n.

A186236 G.f.: exp( Sum_{n>=0} [ Sum_{k=0..2*n} A027907(n,k)^2 * x^k ]* x^n/n ), where A027907 is the triangle of trinomial coefficients.

Original entry on oeis.org

1, 1, 2, 5, 13, 34, 93, 262, 753, 2198, 6502, 19449, 58724, 178739, 547836, 1689407, 5237939, 16318137, 51056027, 160363129, 505456920, 1598263936, 5068483189, 16116397411, 51371962474, 164123564499, 525447953073, 1685534207788, 5416719384326, 17437073203711
Offset: 0

Views

Author

Paul D. Hanna, Oct 19 2011

Keywords

Comments

Trinomial coefficients satisfy: Sum_{k=0..2*n} A027907(n,k)*x^k = (1+x+x^2)^n.

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 13*x^4 + 34*x^5 + 93*x^6 +...
The logarithm begins:
log(A(x)) = x + 3*x^2/2 + 10*x^3/3 + 31*x^4/4 + 91*x^5/5 + 282*x^6/6 + 890*x^7/7 + 2831*x^8/8 + 9055*x^9/9 + 29133*x^10/10 +...
which equals the sum of the series:
log(A(x)) = (1 + x + x^2)*x
+ (1 + 2^2*x + 3^2*x^2 + 2^2*x^3 + x^4)*x^2/2
+ (1 + 3^2*x + 6^2*x^2 + 7^2*x^3 + 6^2*x^4 + 3*x^5 + x^6)*x^3/3
+ (1 + 4^2*x + 10^2*x^2 + 16^2*x^3 + 19^2*x^4 + 16^2*x^5 + 10^2*x^6 + 4^2*x^7 + x^8)*x^4/4
+ (1 + 5^2*x + 15^2*x^2 + 30^2*x^3 + 45^2*x^4 + 51^2*x^5 + 45^2*x^6 + 30^2*x^7 + 15^2*x^8 + 5^2*x^9 + x^10)*x^5/5 +...

Crossrefs

Cf. A180718 (variant).

Programs

  • PARI
    {A027907(n,k)=polcoeff((1+x+x^2)^n, k)}
{a(n)=polcoeff(exp(sum(m=1, n, sum(k=0, 2*m, A027907(m,k)^2 *x^k) *x^m/m)+x*O(x^n)), n)}

A180717 G.f.: Sum_{n>=0} [Sum_{k=0..n} C(n,k)^2*x^k]^2 * x^n.

Original entry on oeis.org

1, 1, 3, 10, 37, 140, 544, 2181, 8873, 36647, 152950, 644313, 2734648, 11681428, 50173541, 216532005, 938383331, 4081653710, 17811999929, 77957939080, 342099306436, 1504801777973, 6633574235109, 29300516237855
Offset: 0

Views

Author

Paul D. Hanna, Sep 29 2010

Keywords

Comments

Compare g.f. to a g.f. of the Whitney numbers (A051286):
. Sum_{n>=0} [Sum_{k=0..n} C(n,k)^2*x^k] * x^n.

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 10*x^3 + 37*x^4 + 140*x^5 + 544*x^6 +...
equals the sum of the series:
A(x) = 1 + (1 + x)^2*x + (1 + 4*x + x^2)^2*x^2
+ (1 + 9*x + 9*x^2 + x^3)^2*x^3
+ (1 + 16*x + 36*x^2 + 16*x^3 + x^4)^2*x^4
+ (1 + 25*x + 100*x^2 + 100*x^3 + 25*x^4 + x^5)^2*x^5
+ (1 + 36*x + 225*x^2 + 400*x^3 + 225*x^4 + 36*x^5 + x^6)^2*x^6 +...

Links

Crossrefs

Cf. A180718.

Programs

  • PARI
    {a(n)=polcoeff(sum(m=0,n,sum(k=0,m,binomial(m,k)^2*x^k)^2*x^m)+x*O(x^n),n)}

Formula

a(n) ~ c * d^n / (Pi*n), where d = 1/3*(5 + (187/2 - (9*sqrt(93))/2)^(1/3) + (1/2*(187 + 9*sqrt(93)))^(1/3)) = 4.61347026758155538... is the root of the equation 1 - 2*d + 5*d^2 - d^3 = 0, c = 1/192*(80 + (382976 - 18432*sqrt(93))^(1/3) + 8*2^(2/3)*(187 + 9*sqrt(93))^(1/3)) = 1.15336756689... is the root of the equation 64*c^3 - 80*c^2 + 8*c - 1 = 0. - Vaclav Kotesovec, Jul 31 2014

A197601 G.f.: exp( Sum_{n>=1} [Sum_{k=0..2*n} C(2*n,k)^2 *x^k] *x^n/n ).

Original entry on oeis.org

1, 1, 5, 14, 52, 187, 708, 2734, 10758, 43004, 174004, 711660, 2936564, 12211688, 51124185, 215299685, 911445413, 3876523626, 16556573129, 70980163570, 305343924258, 1317634326631, 5702146948069, 24741071869651, 107608326588838, 469073933764287
Offset: 0

Views

Author

Paul D. Hanna, Oct 20 2011

Keywords

Examples

			G.f.: A(x) = 1 + x + 5*x^2 + 14*x^3 + 52*x^4 + 187*x^5 + 708*x^6 +...
The logarithm of the g.f. begins:
log(A(x)) = x + 9*x^2/2 + 28*x^3/3 + 121*x^4/4 + 496*x^5/5 + 2100*x^6/6 + 9017*x^7/7 + 38969*x^8/8 +...+ A198059(n)*x^n/n +...
and equals the sum of the series:
log(A(x)) = (1 + 2^2*x + x^2)*x
+ (1 + 4^2*x + 6^2*x^2 + 4^2*x^3 + x^4)*x^2/2
+ (1 + 6^2*x + 15^2*x^2 + 20^2*x^3 + 15^2*x^4 + 6^2*x^5 + x^6)*x^3/3
+ (1 + 8^2*x + 28^2*x^2 + 56^2*x^3 + 70^2*x^4 + 56^2*x^5 + 28^2*x^6 + 8^2*x^7 + x^8)*x^4/4
+ (1 + 10^2*x + 45^2*x^2 + 120^2*x^3 + 210^2*x^4 + 252^2*x^5 + 210^2*x^6 + 120^2*x^7 + 45^2*x^8 + 10^2*x^9 + x^10)*x^5/5 +...
which involves the squares of the coefficients in even powers of (1+x).
The logarithm of the g.f. can also be expressed as:
log(A(x)) = (1-x)^5*(1 + 3^2*x + 6^2*x^2 + 10^2*x^3 + 15^2*x^4 +...)*x
+ (1-x)^9*(1 + 5^2*x + 15^2*x^2 + 35^2*x^3 + 70^2*x^4 +...)*x^2/2
+ (1-x)^13*(1 + 7^2*x + 28^2*x^2 + 84^2*x^3 + 210^2*x^4 +...)*x^3/3
+ (1-x)^17*(1 + 9^2*x + 45^2*x^2 + 165^2*x^3 + 495^2*x^4 +...)*x^4/4
+ (1-x)^21*(1 + 11^2*x + 66^2*x^2 + 286^2*x^3 + 1001^2*x^4 +...)*x^5/5 +...
which involves the squares of the coefficients in odd powers of 1/(1-x).

Links

Crossrefs

Cf. A198059 (log), A186236, A004148, A180717, A180718.

Programs

  • Mathematica
    nmax = 30; CoefficientList[Series[Exp[Sum[Hypergeometric2F1[-2*k, -2*k, 1, x]*x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, May 29 2022 *)
  • PARI
    {a(n)=polcoeff(exp(sum(m=1, n, sum(k=0, n, binomial(2*m,k)^2 *x^k) *x^m/m)+x*O(x^n)), n)}
  • PARI
    {a(n)=polcoeff(exp(sum(m=1, n, (1-x+x*O(x^n))^(4*m+1) *sum(k=0, n-m+1, binomial(2*m+k, k)^2 *x^k+x*O(x^n)) *x^m/m)+x*O(x^n)), n)}

Formula

G.f.: exp( Sum_{n>=1} (1-x)^(4*n+1) *[Sum_{k>=0} C(2*n+k,k)^2 *x^k] *x^n/n ).
Logarithmic derivative equals A198059.

A196559 G.f.: exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^3 * x^k]^2 * x^n/n ).

Original entry on oeis.org

1, 1, 3, 12, 65, 384, 2197, 14078, 94739, 670612, 4899280, 36645899, 281037158, 2197679518, 17489660228, 141241307806, 1155345218645, 9559672712389, 79905432682918, 674005489358155, 5731854529045978, 49105864505432392, 423531623342726441
Offset: 0

Views

Author

Paul D. Hanna, Oct 03 2011

Keywords

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 12*x^3 + 65*x^4 + 384*x^5 + 2197*x^6 +...
where
log(A(x)) = (1 + x)^2*x + (1+2^3*x+x^2)^2*x^2/2 + (1+3^3*x+3^3*x^2+x^3)^2*x^3/3 + (1+4^3*x+6^3*x^2+4^3*x^3+x^4)^2*x^4/4 +...
More explicitly,
log(A(x)) = x + 5*x^2/2 + 28*x^3/3 + 205*x^4/4 + 1506*x^5/5 + 10016*x^6/6 +...

Crossrefs

Cf. A166896, A180718, A196560.

Programs

  • PARI
    {a(n)=polcoeff(exp(sum(m=1, n, sum(k=0, m, binomial(m, k)^3*x^k)^2*x^m/m)+x*O(x^n)), n)}

A196560 G.f.: exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^4 * x^k]^2 * x^n/n ).

Original entry on oeis.org

1, 1, 3, 20, 205, 2624, 24793, 283522, 3639005, 50426826, 740744940, 10801827249, 163698355616, 2554965416964, 40878247859612, 667841855292388, 11051724909284834, 185702751266940874, 3162454792706586691, 54508849210857505845
Offset: 0

Views

Author

Paul D. Hanna, Oct 03 2011

Keywords

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 20*x^3 + 205*x^4 + 2624*x^5 + 24793*x^6 +...
where
log(A(x)) = (1 + x)^2*x + (1+2^4*x+x^2)^2*x^2/2 + (1+3^4*x+3^4*x^2+x^3)^2*x^3/3 + (1+4^4*x+6^4*x^2+4^4*x^3+x^4)^2*x^4/4 +...
More explicitly,
log(A(x)) = x + 5*x^2/2 + 52*x^3/3 + 733*x^4/4 + 11926*x^5/5 + 129944*x^6/6 +...

Crossrefs

Cf. A166898, A180718, A196559.

Programs

  • PARI
    {a(n)=polcoeff(exp(sum(m=1, n, sum(k=0, m, binomial(m, k)^4*x^k)^2*x^m/m)+x*O(x^n)), n)}
Showing 1-6 of 6 results.

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