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 1-8 of 8 results.

A209330 Triangle defined by T(n,k) = binomial(n^2, n*k), for n>=0, k=0..n, as read by rows.

Original entry on oeis.org

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, 1, 85900584, 675248872536, 39049918716424, 39049918716424, 675248872536, 85900584, 1, 1
Offset: 0

Views

Author

Paul D. Hanna, Mar 06 2012

Keywords

Comments

Column 1 equals A014062.
Row sums equal A167009.
Antidiagonal sums equal A209331.
Ignoring initial row T(0,0), equals the logarithmic derivative of the g.f. of triangle A209196.

Examples

			The triangle of coefficients C(n^2,n*k), n>=k, k=0..n, 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;
1, 85900584, 675248872536, 39049918716424, 39049918716424, 675248872536, 85900584, 1; ...

Links

Crossrefs

Cf. A014062 (column 1), A167009 (row sums), A209331, A209196.
Cf. related triangles: A209196 (exp), A228836, A228832, A226234.
Cf. A206830.

Programs

  • Mathematica
    Table[Binomial[n^2, n*k], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Jan 05 2018 *)
  • PARI
    {T(n,k)=binomial(n^2,n*k)}
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))

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),", "))

A209196 Triangle defined by g.f. A(x,y) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n^2, n*k) * y^k ), as read by rows.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 32, 32, 1, 1, 487, 3282, 487, 1, 1, 11113, 657573, 657573, 11113, 1, 1, 335745, 209282906, 1513844855, 209282906, 335745, 1, 1, 12607257, 96673776804, 5580284351032, 5580284351032, 96673776804, 12607257, 1, 1, 565877928, 61162554558200
Offset: 0

Views

Author

Paul D. Hanna, Mar 05 2012

Keywords

Examples

			This triangle begins:
1;
1, 1;
1, 4, 1;
1, 32, 32, 1;
1, 487, 3282, 487, 1;
1, 11113, 657573, 657573, 11113, 1;
1, 335745, 209282906, 1513844855, 209282906, 335745, 1;
1, 12607257, 96673776804, 5580284351032, 5580284351032, 96673776804, 12607257, 1;
1, 565877928, 61162554558200, 31336815578461815, 229089181252258800, 31336815578461815, 61162554558200, 565877928, 1; ...
G.f.: A(x,y) = 1 + (1+y)*x + (1+4*y+y^2)*x^2 + (1+32*y+32*y^2+y^3)*x^3 + (1+487*y+3282*y^2+487*y^3+y^4)*x^4 +...
The logarithm of the g.f. equals the series:
log(A(x,y)) = (1 + y)*x
+ (1 + 6*y + y^2)*x^2/2
+ (1 + 84*y + 84*y^2 + y^3)*x^3/3
+ (1 + 1820*y + 12870*y^2 + 1820*y^3 + y^4)*x^4/4
+ (1 + 53130*y + 3268760*y^2 + 3268760*y^3 + 53130*y^4 + y^5)*x^5/5 +...
in which the coefficients form A209330(n,k) = binomial(n^2, n*k).

Links

Crossrefs

Cf. A209197, A167006, A206830, A209330 (log), A155200.

Programs

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

Formula

Column 1 equals A209197.
Row sums equal A167006.
Antidiagonal sums equal A206830.

A207135 G.f.: exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n^2, k*(n-k)) ).

Original entry on oeis.org

1, 2, 5, 32, 796, 77508, 26058970, 28765221688, 101824384364586, 1145306676113095172, 40618070255705049577152, 4523562146025746408072408406, 1576501611479138389748204925102907, 1714649258669533421310212170714443813118
Offset: 0

Views

Author

Paul D. Hanna, Feb 15 2012

Keywords

Comments

The logarithmic derivative yields A207136.
Equals the row sums of triangle A228900.
Equals the self-convolution of A228852.

Examples

			G.f.: A(x) = 1 + 2*x + 5*x^2 + 32*x^3 + 796*x^4 + 77508*x^5 +...
where the logarithm of the g.f. equals the l.g.f. of A207136:
log(A(x)) = 2*x + 6*x^2/2 + 74*x^3/3 + 2942*x^4/4 + 379502*x^5/5 +...

Crossrefs

Cf. A207136 (log), A167006, A228900, A206830, A206850, A228852.

Programs

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

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),", "))

A228905 G.f.: exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n*k, k^2) * x^k ).

Original entry on oeis.org

1, 1, 2, 3, 5, 12, 33, 139, 1251, 10598, 176642, 4720781, 106779821, 5953841083, 373265833332, 23827795512789, 3914313805097976, 548326897932632059, 108647952177920032693, 45931050219457726501030, 14741338951262398648743248, 9489791738688118291360645939
Offset: 0

Views

Author

Paul D. Hanna, Sep 07 2013

Keywords

Comments

Equals the antidiagonal sums of triangle A228904.

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 12*x^5 + 33*x^6 + 139*x^7 +...
such that, by definition, the logarithm equals (cf. A228832):
log(A(x)) = (1 + x)*x + (1 + 2*x + x^2)*x^2/2 + (1 + 3*x + 15*x^2 + x^3)*x^3/3 + (1 + 4*x + 70*x^2 + 220*x^3 + x^4)*x^4/4 + (1 + 5*x + 210*x^2 + 5005*x^3 + 4845*x^4 + x^5)*x^5/5 +...
More explicitly,
log(A(x)) = x + 3*x^2/2 + 4*x^3/3 + 7*x^4/4 + 31*x^5/5 + 114*x^6/6 + 687*x^7/7 + 8679*x^8/8 + 82948*x^9/9 +...

Links

Crossrefs

Cf. A228904, A228832.
Cf. variants: A206850, A207137, A206830.

Programs

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

A227467 E.g.f.: exp( Sum_{n>=1} (1+x)^(n^2) * x^n/n ).

Original entry on oeis.org

1, 1, 4, 24, 252, 3660, 73560, 1921080, 63411600, 2574406800, 125747475840, 7258472907840, 487590023511360, 37629962101892160, 3299990581104497280, 325758967714868688000, 35904380354917794720000, 4387164775718671231084800, 590610815931660911894707200, 87118296156852814044256665600
Offset: 0

Views

Author

Paul D. Hanna, Aug 24 2013

Keywords

Comments

Compare the definition to: exp( Sum_{n>=1} (1+y)^(n^2) * x^n/n ), which yields an integer series whenever y is an integer.
Note that exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n^2, n*k) * x^k ) yields an integer series (A206830).

Examples

			E.g.f.: A(x) = 1 + x + 4*x^2/2! + 24*x^3/3! + 252*x^4/4! + 3660*x^5/5! +...
where, by definition,
log(A(x)) = (1+x)*x + (1+x)^4*x^2/2 + (1+x)^9*x^3/3 + (1+x)^16*x^4/4 + (1+x)^25*x^5/5+ (1+x)^36*x^6/6+ (1+x)^49*x^7/7 +...

Crossrefs

Cf. A206830, A167006.

Programs

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

A227468 G.f.: exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n^3, n^2*k) * x^k ).

Original entry on oeis.org

1, 1, 2, 37, 1562313, 122131737394518, 26010968765974205465787541, 22347536974721066092798325076069521074882, 113454243067016764816945424312979214671918840299656114590507, 897202601035299299315214220213621062686601174611936477408260666612934393100592315294994
Offset: 0

Views

Author

Paul D. Hanna, Aug 24 2013

Keywords

Comments

Compare the definition to: exp( Sum_{n>=1} (1+y)^(n^3) * x^n/n ), which yields an integer series whenever y is an integer (e.g., A158110).
Note: exp( Sum_{n>=1} (1+x)^(n^3) * x^n/n ) does not yield an integer series.

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 37*x^3 + 1562313*x^4 + 122131737394518*x^5 + ...
such that the logarithm equals
log(A(x)) = (1+x)*x + (1 + 70*x + x^2)*x^2/2
+ (1 + 4686825*x + 4686825*x^2 + x^3)*x^3/3
+ (1 + 488526937079580*x + 1832624140942590534*x^2 + 488526937079580*x^3 + x^4)*x^/4 + ...

Crossrefs

Cf. A158110, A206830, A227467.

Programs

  • PARI
    {a(n)=polcoeff(exp(sum(m=1, n, sum(k=0, m, binomial(m^3, m^2*k)*x^k)*x^m/m)+x*O(x^n)), n)}
for(n=0, 15, print1(a(n), ", "))
Showing 1-8 of 8 results.

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

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