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.

A206849 a(n) = Sum_{k=0..n} binomial(n^2, k^2).

Original entry on oeis.org

1, 2, 6, 137, 13278, 4098627, 8002879629, 66818063663192, 1520456935214867934, 167021181249536494996841, 102867734705055054467692090431, 179314863425920182637610314008444247, 1094998941099523423274757578750950802034789
Offset: 0

Views

Author

Paul D. Hanna, Feb 15 2012

Keywords

Examples

			L.g.f.: L(x) = 2*x + 6*x^2/2 + 137*x^3/3 + 13278*x^4/4 + 4098627*x^5/5 +...
where exponentiation yields the g.f. of A206848:
exp(L(x)) = 1 + 2*x + 5*x^2 + 53*x^3 + 3422*x^4 + 826606*x^5 + 1335470713*x^6 +...
Illustration of terms: by definition,
a(1) = C(1,0) + C(1,1);
a(2) = C(4,0) + C(4,1) + C(4,4);
a(3) = C(9,0) + C(9,1) + C(9,4) + C(9,9);
a(4) = C(16,0) + C(16,1) + C(16,4) + C(16,9) + C(16,16);
a(5) = C(25,0) + C(25,1) + C(25,4) + C(25,9) + C(25,16) + C(25,25);
a(6) = C(36,0) + C(36,1) + C(36,4) + C(36,9) + C(36,16) + C(36,25) + C(36,36); ...
Numerically, the above evaluates to be:
a(1) = 1 + 1 = 2;
a(2) = 1 + 4 + 1 = 6;
a(3) = 1 + 9 + 126 + 1 = 137;
a(4) = 1 + 16 + 1820 + 11440 + 1 = 13278;
a(5) = 1 + 25 + 12650 + 2042975 + 2042975 + 1 = 4098627;
a(6) = 1 + 36 + 58905 + 94143280 + 7307872110 + 600805296 + 1 = 8002879629; ...

Links

Crossrefs

Cf. A206848 (exp), A226234, A206847, A206850, A206851.
Cf. A066382.

Programs

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

Formula

Ignoring the initial term a(0), equals the logarithmic derivative of A206848.
Equals the row sums of triangle A226234.
From Vaclav Kotesovec, Mar 03 2014: (Start)
Limit n->infinity a(n)^(1/n^2) = 2
Lim sup n->infinity a(n)/(2^(n^2)/n) = sqrt(2/Pi) * JacobiTheta3(0,exp(-4)) = Sqrt[2/Pi] * EllipticTheta[3, 0, 1/E^4] = 0.827112271364145742...
Lim inf n->infinity a(n)/(2^(n^2)/n) = sqrt(2/Pi) * JacobiTheta2(0,exp(-4)) = Sqrt[2/Pi] * EllipticTheta[2, 0, 1/E^4] = 0.587247586271786487...
(End)

A206850 G.f.: exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n^2, k^2) * x^k ).

Original entry on oeis.org

1, 1, 2, 4, 8, 56, 522, 5972, 424954, 16560881, 1528544877, 483389731955, 70609119680761, 53933819677734187, 58734216507052608587, 38789122414735365076327, 202547156817505166242299130, 712808848212730366850407506134, 2914935606380176735260119042755221
Offset: 0

Views

Author

Paul D. Hanna, Feb 13 2012

Keywords

Comments

Equals antidiagonal sums of triangle A228902.

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 8*x^4 + 56*x^5 + 522*x^6 + 5972*x^7 +...
such that, by definition, the logarithm equals the series:
log(A(x)) = x*(1+x) + x^2*(1 + 4*x + x^2)/2
+ x^3*(1 + 9*x + 126*x^2 + x^3)/3
+ x^4*(1 + 16*x + 1820*x^2 + 11440*x^3 + x^4)/4
+ x^5*(1 + 25*x + 12650*x^2 + 2042975*x^3 + 2042975*x^4 + x^5)/5
+ x^6*(1 + 36*x + 58905*x^2 + 94143280*x^3 + 7307872110*x^4 + 600805296*x^5 + x^6)/6
+ x^7*(1 + 49*x + 211876*x^2 + 2054455634*x^3 + 3348108992991*x^4 + 63205303218876*x^5 + 262596783764*x^6 + x^7)/7 +...
+ x^n*(Sum_{k=0..n} binomial(n^2, k^2)*x^k)/n  +...

Crossrefs

Cf. A206851 (log), A228902, A206830, A167006.

Programs

  • PARI
    {a(n)=polcoeff(exp(sum(m=1, n, sum(k=0, m, binomial(m^2, k^2)*x^k)*x^m/m)+x*O(x^n)), n)}
for(n=0,25,print1(a(n),", "))

A123165 Row sums of A123163.

Original entry on oeis.org

1, 1, 2, 5, 11, 143, 1847, 24127, 2101931, 96398196, 9362963203, 3376252046640, 551993132054154, 434634824535802596, 528116646162507517308, 372831439174848001477184, 2029862948426766042724907818
Offset: 0

Views

Author

Roger L. Bagula, Oct 02 2006

Keywords

Links

Crossrefs

Cf. A123163, A206851.

Programs

  • Magma
    [(&+[Binomial((n-k)^2, k^2): k in [0..Floor(n/2)]]): n in [0..30]]; // G. C. Greubel, Jul 19 2023
  • Mathematica
    A123163[n_, k_]= ((n-k)^2)!/((k^2)!(n^2-2*n*k)!);
Table[Sum[A123163[n,k], {k,0,n/2}], {n,0,20}]
Table[Sum[Binomial[(n-k)^2,k^2], {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, Mar 04 2014 *)
  • PARI
    {a(n) = sum(k=0, n\2, binomial((n-k)^2, k^2))} \\ Seiichi Manyama, Jan 28 2019
  • SageMath
    def A123165(n): return sum(binomial((n-k)^2, k^2) for k in range(n//2+1))
[A123165(n) for n in range(31)] # G. C. Greubel, Jul 19 2023

Formula

Limit_{n-> oo} a(n)^(1/n^2) = (1-2*r)^r / r^(2*r) = 1.2915356633069917227119166..., where r = A323778 = 0.365498498219858044579736... is the root of the equation (1-r)^(2-2*r) * r^(2*r) = 1-2*r. - Vaclav Kotesovec, Mar 04 2014

Extensions

Edited by N. J. A. Sloane, Oct 04 2006

A206847 a(n) = Sum_{k=0..n} binomial(n^2, k^2) * binomial(n^2, (n-k)^2).

Original entry on oeis.org

1, 2, 18, 2270, 3678482, 51789416252, 9723940840418814, 13783866167176942874214, 260749663122506218247699587346, 35385577627626083328957267246097557212, 64138056102285851525440919122006580387539950268, 814449089808478655249485968539593253265395820497817710866
Offset: 0

Views

Author

Paul D. Hanna, Feb 15 2012

Keywords

Comments

Ignoring the initial term a(0), equals the logarithmic derivative of A206846.

Examples

			L.g.f.: L(x) = 2*x + 18*x^2/2 + 2270*x^3/3 + 3678482*x^4/4 + 51789416252*x^5/5 +...
where exponentiation yields the g.f. of A206846:
exp(L(x)) = 1 + 2*x + 11*x^2 + 776*x^3 + 921193*x^4 + 10359730908*x^5 +...
Illustration of terms: by definition,
a(1) = C(1,0)*C(1,1) + C(1,1)*C(1,0);
a(2) = C(4,0)*C(4,4) + C(4,1)*C(4,1) + C(4,4)*C(4,0);
a(3) = C(9,0)*C(9,9) + C(9,1)*C(9,4) + C(9,4)*C(9,1) + C(9,9)*C(9,0);
a(4) = C(16,0)*C(16,16) + C(16,1)*C(16,9) + C(16,4)*C(16,4) + C(16,9)*C(16,1) + C(16,16)*C(16,0); ...
Numerically, the above evaluates to be:
a(1) = 1*1 + 1*1 = 2;
a(2) = 1*1 + 4*4 + 1*1 = 18;
a(3) = 1*1 + 9*126 + 126*9 + 1*1 = 2270;
a(4) = 1*1 + 16*11440 + 1820*1820 + 11440*16 + 1*1 = 3678482;
a(5) = 1*1 + 25*2042975 + 12650*2042975 + 2042975*12650 + 2042975*25 + 1*1 = 51789416252; ...

Links

Crossrefs

Cf. A206846 (exp), A206849, A206851.

Programs

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

Formula

From Vaclav Kotesovec, Mar 03 2014: (Start)
Limit n->infinity a(n)^(1/n^2) = 16/(3*sqrt(3)).
a(n) ~ c * 2^(4*n^2+3) / (Pi * n^2 * 3^(3*n^2/2+1)), where c = JacobiTheta3(0,9*exp(-16/3)) = EllipticTheta[3, 0, 9*Exp[-16/3]] = 1.08691022925895131... if n is even, and c = JacobiTheta2(0,9*exp(-16/3)) = EllipticTheta[2, 0, 9*Exp[-16/3]] = 0.91485129628884995... if n is odd.
(End)

A323778 Decimal expansion of the root of the equation (1-r)^(2-2*r) * r^(2*r) = 1-2*r.

Original entry on oeis.org

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

Views

Author

Vaclav Kotesovec, Jan 27 2019

Keywords

Examples

			0.36549849821985804457973687544629908832275880696346029501595516768211883674...

Crossrefs

Cf. A123165, A206851.
Cf. A220359, A237421, A245259, A323768, A323773, A323777.

Programs

  • Mathematica
    RealDigits[r/.FindRoot[(1-r)^(2-2*r) * r^(2*r) == 1-2*r, {r, 1/3}, WorkingPrecision->250], 10, 200][[1]]

A207140 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(n^2,k^2).

Original entry on oeis.org

1, 2, 10, 407, 56746, 30771252, 115106662819, 1446405270234360, 53819202633553797290, 12313337704248075967333334, 12373818231445938048765251252260, 33156027144321617106970597265032233270, 409476940913917468665022448013012674533441891
Offset: 0

Views

Author

Paul D. Hanna, Feb 15 2012

Keywords

Comments

Ignoring initial term a(0), equals the logarithmic derivative of A207139.

Examples

			L.g.f.: L(x) = 2*x + 10*x^2/2 + 407*x^3/3 + 56746*x^4/4 + 30771252*x^5/5 +...
where exponentiation equals the g.f. of A207139:
exp(L(x)) = 1 + 2*x + 7*x^2 + 147*x^3 + 14481*x^4 + 6183605*x^5 +...
By definition, the initial terms begin: a(0) = 1;
a(1) = C(1,0)*C(1,0), + C(1,1)*C(1,1);
a(2) = C(2,0)*C(4,0), + C(2,1)*C(4,1), + C(2,2)*C(4,4);
a(3) = C(3,0)*C(9,0), + C(3,1)*C(9,1), + C(3,2)*C(9,4), + C(3,3)*C(9,9);
a(4) = C(4,0)*C(16,0), + C(4,1)*C(16,1), + C(4,2)*C(16,4), + C(4,3)*C(16,9), + C(4,4)*C(16,16); ...
which is evaluated as:
a(1) = 1*1 + 1*1 = 2;
a(2) = 1*1 + 2*4 + 1*1 = 10;
a(3) = 1*1 + 3*9 + 3*126 + 1*1 = 407;
a(4) = 1*1 + 4*16 + 6*1820 + 4*11440 + 1*1 = 56746;
a(5) = 1*1 + 5*25 + 10*12650 + 10*2042975 + 5*2042975 + 1*1 = 30771252;
a(6) = 1*1 + 6*36 + 15*58905 + 20*94143280 + 15*7307872110 + 6*600805296 + 1*1 = 115106662819; ...

Crossrefs

Cf. A207139 (exp), A206851, A207136, A207138, A167009.

Programs

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

Formula

Limit n->infinity a(n)^(1/n^2) = 2. - Vaclav Kotesovec, Mar 03 2014
Showing 1-6 of 6 results.

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

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