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.

A228901 Column 1 of triangle A228900.

Original entry on oeis.org

1, 3, 15, 155, 2685, 65517, 2063205, 79715229, 3653521179, 193876702019, 11696128639195, 790788563411431, 59238930606369007, 4870881096635599567, 436176889680415116607, 42256722956163045301807, 4403938204071514872757477, 491317594015782152377769725, 58423874601431980415156917525
Offset: 1

Views

Author

Paul D. Hanna, Sep 07 2013

Keywords

Comments

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

Crossrefs

Cf. A228900.

Programs

  • PARI
    {a(n)=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), 1, y)}
for(n=1, 20, print1(a(n), ", "))

A228836 Triangle defined by T(n,k) = binomial(n^2, (n-k)*k), for n>=0, k=0..n, as read by rows.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 36, 36, 1, 1, 560, 1820, 560, 1, 1, 12650, 177100, 177100, 12650, 1, 1, 376992, 30260340, 94143280, 30260340, 376992, 1, 1, 13983816, 8217822536, 92263734836, 92263734836, 8217822536, 13983816, 1, 1, 621216192, 3284214703056, 159518999862720, 488526937079580, 159518999862720, 3284214703056, 621216192, 1
Offset: 0

Views

Author

Paul D. Hanna, Sep 05 2013

Keywords

Examples

			The triangle of coefficients C(n^2, (n-k)*k), n>=k, k=0..n, begins:
  1;
  1, 1;
  1, 4, 1;
  1, 36, 36, 1;
  1, 560, 1820, 560, 1;
  1, 12650, 177100, 177100, 12650, 1;
  1, 376992, 30260340, 94143280, 30260340, 376992, 1;
  1, 13983816, 8217822536, 92263734836, 92263734836, 8217822536, 13983816, 1;
  ...

Links

Crossrefs

Cf. A207136 (row sums), A228837 (antidiagonal sums), A070780 (column 1).
Cf. related triangles: A228900(exp), A209330, A226234, A228832.

Programs

  • Mathematica
    T[n_,k_]:=Binomial[n^2, (n-k)*k]; Table[T[n,k],{n,0,8},{k,0,n}]//Flatten (* Stefano Spezia, Aug 02 2025 *)
  • PARI
    {T(n,k)=binomial(n^2, (n-k)*k)}
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))

A228904 Triangle defined by g.f. A(x,y) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n*k, k^2) * y^k ), as read by rows.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 7, 1, 1, 4, 26, 62, 1, 1, 5, 70, 1087, 1031, 1, 1, 6, 155, 9257, 124702, 24782, 1, 1, 7, 301, 51397, 4479983, 26375325, 774180, 1, 1, 8, 532, 215129, 79666708, 5059028293, 8735721640, 29763855, 1, 1, 9, 876, 736410, 891868573, 357346615545, 10783389596184, 4162906254188, 1359654560, 1
Offset: 0

Views

Author

Paul D. Hanna, Sep 07 2013

Keywords

Examples

			This triangle begins:
1;
1, 1;
1, 2, 1;
1, 3, 7, 1;
1, 4, 26, 62, 1;
1, 5, 70, 1087, 1031, 1;
1, 6, 155, 9257, 124702, 24782, 1;
1, 7, 301, 51397, 4479983, 26375325, 774180, 1;
1, 8, 532, 215129, 79666708, 5059028293, 8735721640, 29763855, 1;
1, 9, 876, 736410, 891868573, 357346615545, 10783389596184, 4162906254188, 1359654560, 1;
...
G.f.: A(x,y) = 1 + (1+y)*x + (1+2*y+y^2)*x^2 + (1+3*y+7*y^2+y^3)*x^3 + (1+4*y+26*y^2+62*y^3+y^4)*x^4 + (1+5*y+70*y^2+1087*y^3+1031*y^4+y^5)*x^5 +...
The logarithm of the g.f. equals the series:
log(A(x,y)) = (1 + y)*x
+ (1 + 2*y + y^2)*x^2/2
+ (1 + 3*y + 15*y^2 + y^3)*x^3/3
+ (1 + 4*y + 70*y^2 + 220*y^3 + y^4)*x^4/4
+ (1 + 5*y + 210*y^2 + 5005*y^3 + 4845*y^4 + y^5)*x^5/5
+ (1 + 6*y + 495*y^2 + 48620*y^3 + 735471*y^4 + 142506*y^5 + y^6)*x^6/6 +...
in which the coefficients form A228832(n,k) = binomial(n*k, k^2).

Crossrefs

Cf. A228809 (row sums), A228905 (antidiagonal sums), A228906 (diagonal).
Cf. related triangles: A228832 (log), A209196, A228900, A228902.

Programs

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

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),", "))

A228902 Triangle defined by g.f. A(x,y) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n^2, k^2) * y^k ), as read by rows.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 6, 45, 1, 1, 10, 505, 2905, 1, 1, 15, 3045, 412044, 411500, 1, 1, 21, 12880, 16106168, 1218805926, 100545716, 1, 1, 28, 43176, 309616264, 479536629727, 9030648908720, 37614371968, 1, 1, 36, 122640, 3752248896, 61545730104024, 50139332516318674, 139855355007409180, 19977489354808, 1
Offset: 0

Views

Author

Paul D. Hanna, Sep 07 2013

Keywords

Examples

			This triangle begins:
  1;
  1, 1;
  1, 3, 1;
  1, 6, 45, 1;
  1, 10, 505, 2905, 1;
  1, 15, 3045, 412044, 411500, 1;
  1, 21, 12880, 16106168, 1218805926, 100545716, 1;
  1, 28, 43176, 309616264, 479536629727, 9030648908720, 37614371968, 1;
  1, 36, 122640, 3752248896, 61545730104024, 50139332516318674, 139855355007409180, 19977489354808, 1;
  ...
G.f.: A(x,y) = 1 + (1+y)*x + (1+3*y+y^2)*x^2 + (1+6*y+45*y^2+y^3)*x^3 + (1+10*y+505*y^2+2905*y^3+y^4)*x^4 + (1+15*y+3045*y^2+412044*y^3+411500*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 + 9*y + 126*y^2 + y^3)*x^3/3
   + (1 + 16*y + 1820*y^2 + 11440*y^3 + y^4)*x^4/4
   + (1 + 25*y + 12650*y^2 + 2042975*y^3 + 2042975*y^4 + y^5)*x^5/5
   + (1 + 36*y + 58905*y^2 + 94143280*y^3 + 7307872110*y^4 + 600805296*y^5 + y^6)*x^/6
   + ...
in which the coefficients form A226234(n,k) = binomial(n^2, k^2).

Crossrefs

Cf. A206848 (row sums), A206850 (antidiagonal sums), A228903 (diagonal).
Cf. related triangles: A226234 (log), A209196, A228900, A228904.

Programs

  • PARI
    {T(n, k)=polcoeff(polcoeff(exp(sum(m=1, n, x^m/m*sum(j=0, m, binomial(m^2, j^2)*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),", "))
Showing 1-6 of 6 results.

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

Content is available under The OEIS End-User License Agreement.