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 1-10 of 16 results. Next

A181143 G.f.: A(x,y) = exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^3*y^k] * x^n/n ) = Sum_{n>=0,k=0..n} T(n,k)*x^n*y^k, as a triangle of coefficients T(n,k) read by rows.

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 1, 14, 14, 1, 1, 30, 85, 30, 1, 1, 55, 337, 337, 55, 1, 1, 91, 1029, 2230, 1029, 91, 1, 1, 140, 2632, 10549, 10549, 2632, 140, 1, 1, 204, 5922, 39533, 73157, 39533, 5922, 204, 1, 1, 285, 12090, 124805, 384948, 384948, 124805, 12090, 285, 1, 1
Offset: 0

Views

Author

Paul D. Hanna, Oct 13 2010

Keywords

Comments

Compare g.f. to that of the following triangle variants:
* Pascal's: exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)*y^k] * x^n/n );
* Narayana: exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^2*y^k] * x^n/n );
* A181144: exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^4*y^k] * x^n/n );
* A218115: exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^5*y^k] * x^n/n );
* A218116: exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^6*y^k] * x^n/n ).

Examples

			G.f.: A(x,y) = 1 + (1+y)*x + (1+5*y+y^2)*x^2 + (1+14*y+14*y^2+y^3)*x^3 + (1+30*y+85*y^2+30*y^3+y^4)*x^4 +...
The logarithm of the g.f. equals the series:
log(A(x,y)) = (1 + y)*x
+ (1 + 2^3*y + y^2)*x^2/2
+ (1 + 3^3*y + 3^3*y^2 + y^3)*x^3/3
+ (1 + 4^3*y + 6^3*y^2 + 4^3*y^3 + y^4)*x^4/4
+ (1 + 5^3*y + 10^3*y^2 + 10^3*y^3 + 5^3*y^4 + y^5)*x^5/5 +...
Triangle begins:
1;
1, 1;
1, 5, 1;
1, 14, 14, 1;
1, 30, 85, 30, 1;
1, 55, 337, 337, 55, 1;
1, 91, 1029, 2230, 1029, 91, 1;
1, 140, 2632, 10549, 10549, 2632, 140, 1;
1, 204, 5922, 39533, 73157, 39533, 5922, 204, 1;
1, 285, 12090, 124805, 384948, 384948, 124805, 12090, 285, 1;
1, 385, 22869, 345389, 1648478, 2748240, 1648478, 345389, 22869, 385, 1;
1, 506, 40678, 861080, 6016297, 15525056, 15525056, 6016297, 861080, 40678, 506, 1; ...
Note that column 1 forms the sum of squares (A000330).
Inverse binomial transform of columns begins:
[1];
[1, 4, 5, 2];
[1, 13, 58, 123, 136, 76, 17];
[1, 29, 278, 1308, 3532, 5867, 6118, 3914, 1407, 218];
[1, 54, 920, 7626, 36916, 114637, 240271, 348354, 350881, 241531, 108551, 28742, 3404]; ...
the g.f. of the rightmost coefficients of which form the g.f. exp( Sum_{n>=1} (3*n)!/(3*n!^3) * x^n/n ), and yield the self-convolution of A229452.

Crossrefs

Cf. A000330 (column 1), A166990 (row sums), A166896 (antidiagonal sums), A218139.
Cf. variants: A001263 (Narayana), A181144, A218115, A218116.

Programs

  • PARI
    {T(n,k)=polcoeff(polcoeff(exp(sum(m=1,n,sum(j=0,m,binomial(m,j)^3*y^j)*x^m/m)+O(x^(n+1))),n,x),k,y)}
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))

A166991 G.f.: A(x) = exp( Sum_{n>=1} A000172(n)*x^n/(2*n) ) where Franel number A000172(n) = Sum_{k=0..n} C(n,k)^3.

Original entry on oeis.org

1, 1, 3, 12, 57, 300, 1693, 10045, 61890, 392688, 2550843, 16891566, 113660475, 775223595, 5349057132, 37280705406, 262119009927, 1857241951359, 13250054817027, 95110710932424, 686490953423700, 4979704242810870
Offset: 0

Views

Author

Paul D. Hanna, Nov 17 2009

Keywords

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 12*x^3 + 57*x^4 + 300*x^5 + 1693*x^6 +...
log(A(x)^2) = 2*x + 10*x^2/2 + 56*x^3/3 + 346*x^4/4 + 2252*x^5/5 + 15184*x^6/6 + 104960*x^7/7 +...+ A000172(n)*x^n/n +...

Links

Crossrefs

Cf. A000172 (Franel numbers), A166990, A166993, A218118, A218120.

Programs

  • Mathematica
    a[n_] := Sum[(Binomial[n, k])^3, {k, 0, n}]; f[x_] := Sum[a[n]*x^n/(2*n), {n, 1, 75}]; CoefficientList[Series[Exp[f[x]], {x, 0, 50}], x] (* G. C. Greubel, May 30 2016 *)
  • PARI
    {a(n)=polcoeff(exp(sum(m=1,n,sum(k=0,m,binomial(m,k)^3)/2*x^m/m)+x*O(x^n)),n)}

Formula

Self-convolution yields A166990.
a(n) ~ c * 8^n / n^2, where c = 0.231776... - Vaclav Kotesovec, Nov 27 2017

A166992 G.f.: A(x) = exp( Sum_{n>=1} A005260(n)*x^n/n ) where A005260(n) = Sum_{k=0..n} C(n,k)^4.

Original entry on oeis.org

1, 2, 11, 74, 621, 5850, 60212, 659712, 7583514, 90494068, 1112755389, 14022849582, 180362150901, 2360201899690, 31344689243344, 421621652965160, 5734850816825046, 78773961705345324, 1091497852618784390
Offset: 0

Views

Author

Paul D. Hanna, Nov 17 2009

Keywords

Examples

			G.f.: A(x) = 1 + 2*x + 11*x^2 + 74*x^3 + 621*x^4 + 5850*x^5 + 60212*x^6 +...
log(A(x)) = 2*x + 18*x^2/2 + 164*x^3/3 + 1810*x^4/4 + 21252*x^5/5 + 263844*x^6/6 + 3395016*x^7/7 +...+ A005260(n)*x^n/n +...

Links

Crossrefs

Cf. A005260, A166990, A166993.

Programs

  • Mathematica
    a[n_] := Sum[(Binomial[n, k])^4, {k, 0, n}]; f[x_] := Sum[a[n]*x^n/(n), {n, 1, 75}]; CoefficientList[Series[Exp[f[x]], {x, 0, 50}], x] (* G. C. Greubel, May 30 2016 *)
  • PARI
    {a(n)=polcoeff(exp(sum(m=1,n,sum(k=0,m,binomial(m,k)^4)*x^m/m)+x*O(x^n)),n)}

Formula

Self-convolution of A166993.
a(n) ~ c * 16^n / n^(5/2), where c = 0.30919827904959014083681667605470681109347914449671378054261267779... - Vaclav Kotesovec, Nov 27 2017

A216354 G.f.: A(x) = exp( Sum_{n>=1} A000172(n)^n*x^n/n ) where Franel number A000172(n) = Sum_{k=0..n} C(n,k)^3.

Original entry on oeis.org

1, 2, 52, 58640, 3583098592, 11584364000042912, 2042518153012624794424576, 20047892010468651075834167466942080, 11138509206681372983092694151616405935206616064, 354938139483847646086359348765071470756626699510545192807936
Offset: 0

Views

Author

Paul D. Hanna, Sep 04 2012

Keywords

Examples

			G.f.: A(x) = 1 + 2*x + 52*x^2 + 58640*x^3 + 3583098592*x^4 +...
where
log(A(x)) = 2*x + 10^2*x^2/2 + 56^3*x^3/3 + 346^4*x^4/4 + 2252^5*x^5/5 + 15184^6*x^6/6 + 104960^7*x^7/7 +...+ A000172(n)^n*x^n/n +...

Crossrefs

Cf. A216355, A166990, A216352, A216353, A000172.

Programs

  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, sum(j=0, m, binomial(m, j)^3)^m*x^m/m+x*O(x^n)))); polcoeff(A, n)}
for(n=0, 31, print1(a(n), ", "))

A216352 G.f.: A(x) = exp( Sum_{n>=1} A000172(n)^2*x^n/n ) where Franel number A000172(n) = Sum_{k=0..n} C(n,k)^3.

Original entry on oeis.org

1, 4, 58, 1256, 35771, 1200188, 45016678, 1827941560, 78753548245, 3551810922324, 166120394053698, 8002733850225288, 395089619067741926, 19911864121386482264, 1021345223473335336668, 53190166903606336969840, 2807000233813092463820488, 149869216802426305919295328
Offset: 0

Views

Author

Paul D. Hanna, Sep 04 2012

Keywords

Examples

			G.f.: A(x) = 1 + 4*x + 58*x^2 + 1256*x^3 + 35771*x^4 + 1200188*x^5 +...
such that
log(A(x)) = 4*x + 100*x^2/2 + 3136*x^3/3 + 119716*x^4/4 + 5071504*x^5/5 +...+ A000172(n)^2*x^n/n +...

Crossrefs

Cf. A166990, A216353, A216354, A216355, A052144, A000172.

Programs

  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, sum(j=0, m, binomial(m, j)^3)^2*x^m*1^m/m+x*O(x^n)))); polcoeff(A, n)}
for(n=0, 31, print1(a(n), ", "))

A216353 G.f.: A(x) = exp( Sum_{n>=1} A000172(n)^3*x^n/n ) where Franel number A000172(n) = Sum_{k=0..n} C(n,k)^3.

Original entry on oeis.org

1, 8, 532, 62624, 10964914, 2399234384, 609215149096, 171739556144192, 52316948995446679, 16918106849112020088, 5736533516906891508780, 2021549577502367744673888, 735516733692051220039803750, 274907827442478316252748869104, 105138174536582510069969443280760
Offset: 0

Views

Author

Paul D. Hanna, Sep 04 2012

Keywords

Examples

			G.f.: A(x) = 1 + 8*x + 532*x^2 + 62624*x^3 + 10964914*x^4 + 2399234384*x^5 +...
where
log(A(x)) = 2^3*x + 10^3*x^2/2 + 56^3*x^3/3 + 346^3*x^4/4 + 2252^3*x^5/5 + 15184^3*x^6/6 + 104960^3*x^7/7 +...+ A000172(n)^3*x^n/n +...

Crossrefs

Cf. A166990, A216352, A216354, A216355, A052144, A000172.

Programs

  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, sum(j=0, m, binomial(m, j)^3)^3*x^m*1^m/m+x*O(x^n)))); polcoeff(A, n)}
for(n=0, 31, print1(a(n), ", "))

A216355 G.f.: A(x) = exp( Sum_{n>=1} A000172(n^2)*x^n/n ) where Franel number A000172(n) = Sum_{k=0..n} C(n,k)^3.

Original entry on oeis.org

1, 2, 175, 1760658, 1583078442003, 109611485085305859618, 547114144500297420116784959134, 189879050147329004652707990280499398833960, 4482752989702739533106941067588051779825642693578987967, 7097288803262045586874332782527584396862908242415791224663533782367102
Offset: 0

Views

Author

Paul D. Hanna, Sep 04 2012

Keywords

Examples

			G.f.: A(x) = 1 + 2*x + 175*x^2 + 1760658*x^3 + 1583078442003*x^4 +...
where
log(A(x)) = 2*x + 346*x^2/2 + 5280932*x^3/3 + 6332299624282*x^4/4 + 548057409594239814752*x^5/5 +...+ A000172(n^2)*x^n/n +...

Crossrefs

Cf. A216356, A166990, A216352, A216353, A216354, A000172.

Programs

  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, sum(j=0, m^2, binomial(m^2, j)^3)*x^m/m+x*O(x^n)))); polcoeff(A, n)}
for(n=0, 31, print1(a(n), ", "))

A218117 G.f.: A(x) = exp( Sum_{n>=1} A005261(n)*x^n/n ) where A005261(n) = Sum_{k=0..n} C(n,k)^5.

Original entry on oeis.org

1, 2, 19, 198, 2961, 49566, 938322, 19083624, 412160478, 9305822076, 217855152321, 5251363667622, 129704365956114, 3269927116717728, 83893626609970281, 2185188966488265718, 57673989852987800966, 1539973309401567102832, 41544812360973818992909
Offset: 0

Views

Author

Paul D. Hanna, Oct 21 2012

Keywords

Comments

Compare to a g.f. of Catalan numbers (A000108):
exp( Sum_{n>=1} A000984(n)*x^n/n ) where A000984(n) = Sum_{k=0..n} C(n,k)^2.

Examples

			G.f.: A(x) = 1 + 2*x + 19*x^2 + 198*x^3 + 2961*x^4 + 49566*x^5 + 938322*x^6 +...
log(A(x)) = 2*x + 34*x^2/2 + 488*x^3/3 + 9826*x^4/4 + 206252*x^5/5 + 4734304*x^6/6 + 113245568*x^7/7 +...+ A005261(n)*x^n/n +...

Crossrefs

Cf. A218115, A218118, A166990, A166992, A218119, A005261.

Programs

  • PARI
    {a(n)=polcoeff(exp(sum(m=1, n, sum(k=0, m, binomial(m, k)^5)*x^m/m)+x*O(x^n)), n)}
for(n=0,25,print1(a(n),", "))

Formula

Equals row sums of triangle A218115.
Self-convolution of A218118.

A218119 G.f.: A(x) = exp( Sum_{n>=1} A069865(n)*x^n/n ) where A069865(n) = Sum_{k=0..n} C(n,k)^6.

Original entry on oeis.org

1, 2, 35, 554, 15297, 451842, 15929824, 601077640, 24488754772, 1046792248856, 46718718597567, 2155032002133834, 102259392504591235, 4967499746642163574, 246231868462969357492, 12419324761881256326288, 635990044563649443993091, 33006906229799699591298070
Offset: 0

Views

Author

Paul D. Hanna, Oct 21 2012

Keywords

Comments

Compare to a g.f. of Catalan numbers (A000108):
exp( Sum_{n>=1} A000984(n)*x^n/n ) where A000984(n) = Sum_{k=0..n} C(n,k)^2.

Examples

			G.f.: A(x) = 1 + 2*x + 35*x^2 + 554*x^3 + 15297*x^4 + 451842*x^5 + 15929824*x^6 +...
log(A(x)) = 2*x + 66*x^2/2 + 1460*x^3/3 + 54850*x^4/4 + 2031252*x^5/5 + 86874564*x^6/6 + 3848298792*x^7/7 +...+ A069865(n)*x^n/n +...

Crossrefs

Cf. A218116, A218120, A166990, A166992, A218117, A069865.

Programs

  • PARI
    {a(n)=polcoeff(exp(sum(m=1, n, sum(k=0, m, binomial(m, k)^6)*x^m/m)+x*O(x^n)), n)}
for(n=0,25,print1(a(n),", "))

Formula

Equals row sums of triangle A218116.
Self-convolution of A218120.

A242903 G.f. A(x) satisfies: coefficient of x^n in A(x)^(2*n) equals A000172(n) = Sum_{k=0..n} C(n,k)^3, the n-th Franel number.

Original entry on oeis.org

1, 1, 1, 1, 3, 8, 26, 89, 324, 1225, 4786, 19170, 78408, 326275, 1377772, 5891401, 25467509, 111144579, 489145720, 2168854885, 9681072845, 43473716527, 196286934526, 890640262188, 4059500301390, 18579693200838, 85360357637580, 393548515741979, 1820335724153452, 8445294476235727, 39291407672079211
Offset: 0

Views

Author

Paul D. Hanna, May 25 2014

Keywords

Examples

			G.f.: A(x) = 1 + x + x^2 + x^3 + 3*x^4 + 8*x^5 + 26*x^6 + 89*x^7 + 324*x^8 +...
Form a table of coefficients in A(x)^(2*n) as follows:
[1,  0,   0,    0,    0,     0,      0,      0,       0,       0, ...];
[1,  2,   3,    4,    9,    24,     75,    252,     903,    3376, ...];
[1,  4,  10,   20,   43,   108,    316,   1020,    3537,   12908, ...];
[1,  6,  21,   56,  138,   354,   1002,   3120,   10485,   37318, ...];
[1,  8,  36,  120,  346,   960,   2756,   8448,   27723,   96440, ...];
[1, 10,  55,  220,  735,  2252,   6785,  21020,   68340,  233870, ...];
[1, 12,  78,  364, 1389,  4716,  15184,  48588,  159186,  541424, ...];
[1, 14, 105,  560, 2408,  9030,  31304, 104960,  351792, 1203244, ...];
[1, 16, 136,  816, 3908, 16096,  60184, 213152,  739162, 2570464, ...];
[1, 18, 171, 1140, 6021, 27072, 109047, 409500, 1480293, 5280932, ...]; ...
then the main diagonal forms the Franel numbers:
[1, 2, 10, 56, 346, 2252, 15184, 104960, 739162, 5280932, ...].

Crossrefs

Cf. A166990, A088220, A000172.

Programs

  • PARI
    {a(n)=polcoeff(sqrt(x/serreverse(x*exp(sum(m=1, n+1, sum(k=0, m, binomial(m, k)^3)*x^m/m +x^2*O(x^n))))),n)}
for(n=0,30,print1(a(n),", "))

Formula

G.f.: sqrt( x / Series_Reversion( x*exp( Sum_{n>=1} A000172(n)*x^n/n ) ) ), where A000172(n) is the n-th Franel number.
[x^n] A(x)^(2*n+2) = (n+1)*A166990(n).
Convolution square-root of A088220.
Showing 1-10 of 16 results. Next

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