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.

A206151 G.f.: exp( Sum_{n>=1} A206152(n)*x^n/n ), where A206152(n) = Sum_{k=0..n} binomial(n,k)^(n+k).

Original entry on oeis.org

1, 2, 7, 120, 16257, 22426576, 181974299842, 15238138790731690, 8413234043413844801094, 36597622942948070873495055416, 1557743574279376981523155294991683637, 377269728353963189455845962558983304322979834
Offset: 0

Views

Author

Paul D. Hanna, Feb 04 2012

Keywords

Comments

Logarithmic derivative yields A206152.

Examples

			G.f.: A(x) = 1 + 2*x + 7*x^2 + 120*x^3 + 16257*x^4 + 22426576*x^5 +...
where the logarithm of the g.f. begins:
log(A(x)) = 2*x + 10*x^2/2 + 326*x^3/3 + 64066*x^4/4 + 111968752*x^5/5 +...+ A206152(n)*x^n/n +...

Crossrefs

Cf. A206152 (log), A184730.

Programs

  • PARI
    {a(n)=polcoeff(exp(sum(m=1,n+1,x^m/m*sum(k=0,m,binomial(m,k)^(m+k))+x*O(x^n))),n)}
for(n=0,16,print1(a(n),", "))

A237421 Decimal expansion of the root of the equation (1-r)^(2*r) = r^(2*r+1).

Original entry on oeis.org

6, 0, 3, 2, 3, 2, 6, 8, 3, 7, 7, 4, 1, 3, 6, 2, 0, 6, 2, 2, 0, 1, 9, 2, 6, 5, 0, 9, 4, 8, 6, 6, 8, 2, 2, 0, 4, 2, 0, 9, 6, 2, 5, 1, 4, 2, 1, 1, 7, 4, 8, 7, 4, 8, 0, 4, 2, 4, 0, 3, 6, 1, 5, 4, 8, 8, 9, 6, 6, 5, 7, 3, 0, 6, 0, 4, 3, 0, 0, 6, 9, 7, 3, 8, 3, 0, 3, 2, 1, 9, 2, 7, 2, 3, 5, 2, 6, 2, 5, 4, 7, 6, 9, 7, 7, 3
Offset: 0

Views

Author

Vaclav Kotesovec, Mar 04 2014

Keywords

Examples

			0.6032326837741362...

Links

Crossrefs

Cf. A206152, A227403, A220359.

Programs

  • Maple
    Digits:= 140:
v:= convert(fsolve((1-r)^(2*r) = r^(2*r+1), r=1/2), string):
seq(parse(v[n+2]), n=0..120);
  • Mathematica
    RealDigits[r/.FindRoot[(1-r)^(2*r)==r^(2*r+1), {r, 1/2}, WorkingPrecision->250], 10, 200][[1]]

A227403 a(n) = Sum_{k=0..n} binomial(n^2, n*k) * binomial(n*k, k^2).

Original entry on oeis.org

1, 2, 14, 1514, 1308582, 17304263902, 1362702892177706, 1323407909279927430346, 11218363871234340925730020646, 637467717878006909442727527733810142, 519660435252919757259949810325837093364580014, 2289503386759572781844843312201361014103189493095636611
Offset: 0

Views

Author

Paul D. Hanna, Sep 20 2013

Keywords

Examples

			The following triangles illustrate the terms involved in the sum
a(n) = Sum_{k=0..n} A209330(n,k) * A228832(n,k).
Triangle A209330(n,k) = binomial(n^2, n*k) begins:
1;
1, 1;
1, 6, 1;
1, 84, 84, 1;
1, 1820, 12870, 1820, 1;
1, 53130, 3268760, 3268760, 53130, 1;
1, 1947792, 1251677700, 9075135300, 1251677700, 1947792, 1;
...
Triangle A228832(n,k) = binomial(n*k, k^2) begins:
1;
1, 1;
1, 2, 1;
1, 3, 15, 1;
1, 4, 70, 220, 1;
1, 5, 210, 5005, 4845, 1;
1, 6, 495, 48620, 735471, 142506, 1; ...

Links

Crossrefs

Cf. A209330, A228832, A206849, A206152, A237421.

Programs

  • Mathematica
    Table[Sum[Binomial[n^2,n*k]*Binomial[n*k,k^2],{k,0,n}],{n,0,10}] (* Vaclav Kotesovec, Sep 21 2013 *)
r^(-(1+r)^2/(2*r))/.FindRoot[(1-r)^(2*r) == r^(2*r+1), {r,1/2}, WorkingPrecision->50] (* program for numerical value of the limit n->infinity a(n)^(1/n^2), Vaclav Kotesovec, Sep 21 2013 *)
  • PARI
    {a(n)=sum(k=0, n, binomial(n^2, n*k)*binomial(n*k, k^2))}
for(n=0,20,print1(a(n),", "))

Formula

a(n) = Sum_{k=0..n} (n^2)! / ( (n^2-n*k)! * (n*k-k^2)! * (k^2)! ).
Limit n->infinity a(n)^(1/n^2) = r^(-(1+r)^2/(2*r)) = 2.93544172048274..., where r = 0.6032326837741362... (see A237421) is the root of the equation (1-r)^(2*r) = r^(2*r+1). - Vaclav Kotesovec, Sep 21 2013

A206154 a(n) = Sum_{k=0..n} binomial(n,k)^(k+2).

Original entry on oeis.org

1, 2, 10, 110, 2386, 125752, 14921404, 3697835668, 2223231412546, 3088517564289836, 9040739066816429380, 63462297965044771663708, 1064766030857977088480630740, 37863276208844960432962611293828, 3144384748384240804260912067907833280
Offset: 0

Views

Author

Paul D. Hanna, Feb 04 2012

Keywords

Comments

Ignoring initial term a(0), equals the logarithmic derivative of A206153.

Examples

			L.g.f.: L(x) = 2*x + 10*x^2/2 + 110*x^3/3 + 2386*x^4/4 + 125752*x^5/5 +...
where exponentiation yields A206151:
exp(L(x)) = 1 + 2*x + 7*x^2 + 48*x^3 + 693*x^4 + 26632*x^5 + 2542514*x^6 +...
Illustration of initial terms:
a(1) = 1^2 + 1^3 = 2;
a(2) = 1^2 + 2^3 + 1^4 = 10;
a(3) = 1^2 + 3^3 + 3^4 + 1^5 = 110;
a(4) = 1^2 + 4^3 + 6^4 + 4^5 + 1^6 = 2386;
a(5) = 1^2 + 5^3 + 10^4 + 10^5 + 5^6 + 1^7 = 125752; ...

Crossrefs

Cf. A206153 (exp), A184731, A206156, A206158, A206152, A167008, A220359.

Programs

  • Mathematica
    Table[Sum[Binomial[n,k]^(k+2),{k,0,n}],{n,0,20}] (* Harvey P. Dale, Jan 16 2014 *)
  • PARI
    {a(n)=sum(k=0,n,binomial(n,k)^(k+2))}
for(n=0,16,print1(a(n),", "))

Formula

Limit n->infinity a(n)^(1/n^2) = (1-r)^(-r/2) = 1.53362806511..., where r = 0.70350607643... (see A220359) is the root of the equation (1-r)^(2*r-1) = r^(2*r). - Vaclav Kotesovec, Jan 29 2014

A206156 a(n) = Sum_{k=0..n} binomial(n,k)^(2*k).

Original entry on oeis.org

1, 2, 6, 92, 5410, 1400652, 2687407464, 18947436116184, 536104663173431874, 130559883231879141946580, 136031455187223511721647272376, 483565526783420050082035900177878504, 14487924180895151383693101563813954330590756
Offset: 0

Views

Author

Paul D. Hanna, Feb 04 2012

Keywords

Comments

Ignoring initial term a(0), equals the logarithmic derivative of A206155.

Examples

			L.g.f.: L(x) = 2*x + 6*x^2/2 + 92*x^3/3 + 5410*x^4/4 + 1400652*x^5/5 +...
where exponentiation yields A206155:
exp(L(x)) = 1 + 2*x + 5*x^2 + 38*x^3 + 1425*x^4 + 283002*x^5 + 448468978*x^6 +...
Illustration of initial terms:
a(1) = 1^0 + 1^2 = 2;
a(2) = 1^0 + 2^2 + 1^4 = 6;
a(3) = 1^0 + 3^2 + 3^4 + 1^6 = 92;
a(4) = 1^0 + 4^2 + 6^4 + 4^6 + 1^8 = 5410;
a(5) = 1^0 + 5^2 + 10^4 + 10^6 + 5^8 + 1^10 = 1400652; ...

Crossrefs

Cf. A206155 (exp), A184731, A206154, A206158, A206152, A220359.

Programs

  • Mathematica
    Table[Sum[Binomial[n,k]^(2*k), {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Mar 03 2014 *)
  • PARI
    {a(n)=sum(k=0,n,binomial(n,k)^(2*k))}
for(n=0,16,print1(a(n),", "))

Formula

Limit n->infinity a(n)^(1/n^2) = r^(2*r^2/(1-2*r)) = 2.3520150420944489879258119..., where r = 0.70350607643066243... (see A220359) is the root of the equation (1-r)^(2*r-1) = r^(2*r). - Vaclav Kotesovec, Mar 03 2014

A206158 a(n) = Sum_{k=0..n} binomial(n,k)^(2*k+1).

Original entry on oeis.org

1, 2, 10, 272, 24226, 12053252, 40086916024, 429254371605824, 23527609330364490754, 10714627376371224032350052, 16964729291782419425708732425300, 109783535843179466164398767001178968704, 6782057095273243388704415924996348722446049600
Offset: 0

Views

Author

Paul D. Hanna, Feb 04 2012

Keywords

Comments

Ignoring initial term a(0), equals the logarithmic derivative of A206157.

Examples

			L.g.f.: L(x) = 2*x + 10*x^2/2 + 272*x^3/3 + 24226*x^4/4 + 12053252*x^5/5 +...
where exponentiation yields A206157:
exp(L(x)) = 1 + 2*x + 7*x^2 + 102*x^3 + 6261*x^4 + 2423430*x^5 + 6686021554*x^6 +...
Illustration of initial terms:
a(1) = 1^1 + 1^3 = 2;
a(2) = 1^1 + 2^3 + 1^5 = 10;
a(3) = 1^1 + 3^3 + 3^5 + 1^7 = 272;
a(4) = 1^1 + 4^3 + 6^5 + 4^7 + 1^9 = 24226;
a(5) = 1^1 + 5^3 + 10^5 + 10^7 + 5^9 + 1^11 = 12053252; ...

Crossrefs

Cf. A206157 (exp), A184731, A206154, A206156, A206152, A220359.

Programs

  • Mathematica
    Table[Sum[Binomial[n,k]^(2*k+1), {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Mar 03 2014 *)
  • PARI
    {a(n)=sum(k=0,n,binomial(n,k)^(2*k+1))}
for(n=0,16,print1(a(n),", "))

Formula

Limit n->infinity a(n)^(1/n^2) = r^(2*r^2/(1-2*r)) = 2.3520150420944489879258119..., where r = 0.70350607643066243... (see A220359) is the root of the equation (1-r)^(2*r-1) = r^(2*r). - Vaclav Kotesovec, Mar 03 2014
Showing 1-6 of 6 results.

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