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.

A207136 a(n) = Sum_{k=0..n} binomial(n^2, k*(n-k)).

Original entry on oeis.org

1, 2, 6, 74, 2942, 379502, 155417946, 200991082378, 814134608643518, 10305926982053248142, 406157795399324680023006, 49758289996116571598723737976, 18917910771770463473290738891259546, 22290399373603219140501180230536732389992
Offset: 0

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Author

Paul D. Hanna, Feb 15 2012

Keywords

Comments

Ignoring initial term a(0), equals the logarithmic derivative of A207135.
Equals the row sums of triangle A228836.

Examples

			L.g.f.: L(x) = 2*x + 6*x^2/2 + 74*x^3/3 + 2942*x^4/4 + 379502*x^5/5 +...
where exponentiation equals the g.f. of A207135:
exp(L(x)) = 1 + 2*x + 5*x^2 + 32*x^3 + 796*x^4 + 77508*x^5 +...
By definition, the initial terms begin: a(0) = 1;
a(1) = C(1,0) + C(1,0);
a(2) = C(4,0) + C(4,1) + C(4,0);
a(3) = C(9,0) + C(9,2) + C(9,2) + C(9,0);
a(4) = C(16,0) + C(16,3) + C(16,4) + C(16,3) + C(16,0);
a(5) = C(25,0) + C(25,4) + C(25,6) + C(25,6) + C(25,4) + C(25,0);
a(6) = C(36,0) + C(36,5) + C(36,8) + C(36,9) + C(36,8) + C(36,5) + C(36,0); ...
which is evaluated as:
a(1) = 1 + 1 = 2;
a(2) = 1 + 4 + 1 = 6;
a(3) = 1 + 36 + 36 + 1 = 74;
a(4) = 1 + 560 + 1820 + 560 + 1 = 2942;
a(5) = 1 + 12650 + 177100 + 177100 + 12650 + 1 = 379502;
a(6) = 1 + 376992 + 30260340 + 94143280 + 30260340 + 376992 + 1 = 155417946; ...

Links

Crossrefs

Cf. A207135 (exp), A167009, A228836.

Programs

  • Maple
    A207136:=n->add(binomial(n^2, k*(n-k)), k=0..n): seq(A207136(n), n=0..15); # Wesley Ivan Hurt, Jun 23 2015
  • Mathematica
    Table[Sum[Binomial[n^2, k*(n-k)],{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Mar 03 2014 *)
  • PARI
    {a(n)=sum(k=0,n,binomial(n^2,(n-k)*k))}
for(n=0,20,print1(a(n),", "))

Formula

a(n) ~ c * 2*sqrt(2/(3*Pi)) * (4/3^(3/4))^(n^2)/n, where c = EllipticTheta[3,0,1/3] = JacobiTheta3(0,1/3) = 1.69145968168171534... if n is even, and c = EllipticTheta[2,0,1/3] = JacobiTheta2(0,1/3) = 1.690611203075214233... if n is odd. - Vaclav Kotesovec, Mar 03 2014

A228900 Triangle defined by g.f. A(x,y) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n^2, (n-k)*k) * y^k ), as read by rows.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 15, 15, 1, 1, 155, 484, 155, 1, 1, 2685, 36068, 36068, 2685, 1, 1, 65517, 5082340, 15763254, 5082340, 65517, 1, 1, 2063205, 1179126560, 13201421078, 13201421078, 1179126560, 2063205, 1, 1, 79715229, 411708127954, 19954261054442, 61092286569334, 19954261054442, 411708127954, 79715229, 1
Offset: 0

Views

Author

Paul D. Hanna, Sep 07 2013

Keywords

Examples

			This triangle begins:
1;
1, 1;
1, 3, 1;
1, 15, 15, 1;
1, 155, 484, 155, 1;
1, 2685, 36068, 36068, 2685, 1;
1, 65517, 5082340, 15763254, 5082340, 65517, 1;
1, 2063205, 1179126560, 13201421078, 13201421078, 1179126560, 2063205, 1;
1, 79715229, 411708127954, 19954261054442, 61092286569334, 19954261054442, 411708127954, 79715229, 1;
...
G.f.: A(x,y) = 1 + (1+y)*x + (1+3*y+y^2)*x^2 + (1+15*y+15*y^2+y^3)*x^3 + (1+155*y+484*y^2+155*y^3+y^4)*x^4 + (1+2685*y+36068*y^2+36068*y^3+2685*y^4+y^5)*x^5 +...
The logarithm of the g.f. equals the series:
log(A(x,y)) = (1 + y)*x
+ (1 + 4*y + y^2)*x^2/2
+ (1 + 36*y + 36*y^2 + y^3)*x^3/3
+ (1 + 560*y + 1820*y^2 + 560*y^3 + y^4)*x^4/4
+ (1 + 12650*y + 177100*y^2 + 177100*y^3 + 12650*y^4 + y^5)*x^5/5
+ (1 + 376992*y + 30260340*y^2 + 94143280*y^3 + 30260340*y^4 + 376992*y^5 + y^6)*x^6/6 +...
in which the coefficients form A228836(n,k) = binomial(n^2, (n-k)*k).

Crossrefs

Cf. A207135 (row sums), A207137 (antidiagonal sums), A228901 (column 1).
Cf. related triangles: A228836 (log), A209196, A228902, A228904.

Programs

  • PARI
    {T(n, k)=polcoeff(polcoeff(exp(sum(m=1, n, x^m/m*sum(j=0, m, binomial(m^2, (m-j)*j)*y^j))+x*O(x^n)), n, x), k, y)}
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))

A207137 G.f.: exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n^2, k*(n-k))*x^k ).

Original entry on oeis.org

1, 1, 2, 4, 17, 171, 3171, 101741, 7181615, 1274607729, 428568152553, 223160743256395, 185627109707405932, 320952534083059792786, 1367454166673309618606950, 11078799748881429582280609036, 137939599816546528357634500253053, 2679390013936303204526656964298150849
Offset: 0

Views

Author

Paul D. Hanna, Feb 15 2012

Keywords

Comments

The logarithmic derivative yields A207138.
Equals the antidiagonal sums of triangle A228900.

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 17*x^4 + 171*x^5 + 3171*x^6 +...
where the logarithm of the g.f. equals the l.g.f. of A207138:
log(A(x)) = x + 3*x^2/2 + 7*x^3/3 + 51*x^4/4 + 761*x^5/5 + 17913*x^6/6 +...

Crossrefs

Cf. A207138 (log), A207135, A228900, A206850, A206830, A167006.

Programs

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

A228809 G.f.: exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n*k, k^2) ).

Original entry on oeis.org

1, 2, 4, 12, 94, 2195, 158904, 31681195, 13904396167, 15305894726347, 44888344014554903, 288228807835914177564, 4270880356112396772814732, 169380654509201278629725097906, 15394658527137259981745081997280638, 3042352591056504014301304188228238554499
Offset: 0

Views

Author

Paul D. Hanna, Sep 04 2013

Keywords

Comments

Logarithmic derivative equals A228808.
Equals row sums of triangle A228904.

Examples

			G.f.: A(x) = 1 + 2*x + 4*x^2 + 12*x^3 + 94*x^4 + 2195*x^5 +...
where
log(A(x)) = 2*x + 4*x^2/2 + 20*x^3/3 + 296*x^4/4 + 10067*x^5/5 + 927100*x^6/6 +...+ A228808(n)*x^n/n +...

Links

Crossrefs

Cf. A228808, A207135, A228904.
Cf. variants: A167006, A206848.

Programs

  • PARI
    {a(n)=polcoeff(exp(sum(m=1, n, x^m/m*sum(k=0, m, binomial(m*k, k^2)))+x*O(x^n)), n)}
for(n=0, 20, print1(a(n), ", "))

A207139 G.f.: exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n,k) * binomial(n^2,k^2) ).

Original entry on oeis.org

1, 2, 7, 147, 14481, 6183605, 19196862399, 206667738393577, 6727813723143519624, 1368162090055314881480420, 1237384559488983889303951699285, 3014186760620644058660289396656407831, 34123084437870355957570087446546456971276065
Offset: 0

Views

Author

Paul D. Hanna, Feb 15 2012

Keywords

Comments

The logarithmic derivative yields A207140.

Examples

			G.f.: A(x) = 1 + 2*x + 7*x^2 + 147*x^3 + 14481*x^4 + 6183605*x^5 +...
where the logarithm of the g.f. equals the l.g.f. of A207140:
log(A(x)) = x + 2*x^2/2 + 10*x^3/3 + 407*x^4/4 + 56746*x^5/5 +...

Crossrefs

Cf. A207140 (log), A206850, A207135, A207137, A167006.

Programs

  • PARI
    {a(n)=polcoeff(exp(sum(m=1,n+1,x^m/m*sum(k=0,m,binomial(m,k)*binomial(m^2,k^2))+x*O(x^n))),n)}
for(n=0,16,print1(a(n),", "))

A228852 G.f.: exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n^2, k*(n-k))/2 ).

Original entry on oeis.org

1, 1, 2, 14, 382, 38344, 12990279, 14369538529, 50897796053428, 572602411324905786, 20308462423438736818782, 2261760763404526386241849803, 788248543938180828988762846368690, 857323841081698966408121705146996762240, 2905542652088907570108828021890682181041282730
Offset: 0

Views

Author

Paul D. Hanna, Sep 05 2013

Keywords

Comments

Self-convolution yields A207135.

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 14*x^3 + 382*x^4 + 38344*x^5 + 12990279*x^6 +...
where
log(A(x)) = x + 3*x^2/2 + 37*x^3/3 + 1471*x^4/4 + 189751*x^5/5 + 77708973*x^6/6 +...+ A207136(n)/2 * x^n/n +...

Crossrefs

Cf. A207135, A207136.

Programs

  • PARI
    {a(n)=polcoeff(exp(sum(m=1, n, x^m/m*sum(k=0, m, binomial(m^2, k*(m-k))/2))+x*O(x^n)), n)}
for(n=0, 20, print1(a(n), ", "))
Showing 1-6 of 6 results.

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