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 1-7 of 7 results.

A209197 Column 1 of triangle A209196.

Original entry on oeis.org

1, 4, 32, 487, 11113, 335745, 12607257, 565877928, 29553415078, 1760584360722, 117828762999498, 8752769915058447, 714626485356930711, 63609663369881873031, 6130647517448380412727, 636052622643842997577302, 70679525819378610579659532, 8375262433274665594692923984
Offset: 1

Views

Author

Paul D. Hanna, Mar 05 2012

Keywords

Comments

G.f. of A209196 is exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n^2, n*k) * y^k ).

Programs

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

A167006 G.f.: exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n^2, n*k) ).

Original entry on oeis.org

1, 2, 6, 66, 4258, 1337374, 1933082159, 11353941470188, 291885138650054688, 29463501750534915665304, 12844314786465829040693498639, 21675661852919288704454219459892060, 156969579902607123047763327413679853875703
Offset: 0

Views

Author

Paul D. Hanna, Nov 17 2009

Keywords

Comments

Logarithmic derivative yields A167009.
Equals row sums of triangle A209196.

Examples

			G.f.: A(x) = 1 + 2*x + 6*x^2 + 66*x^3 + 4258*x^4 + 1337374*x^5 +...
log(A(x)) = 2*x + 8*x^2/2 + 170*x^3/3 + 16512*x^4/4 + 6643782*x^5/5 + 11582386286*x^6/6 +...+ A167009(n)*x^n/n +...

Links

Crossrefs

Cf. A167009, A209196, A155200.
Cf. variants: A206848, A228809.

Programs

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

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(""))

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(""))

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(""))

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(""))

A209424 Triangle defined by g.f.: A(x,y) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n, k)^n * y^k ), as read by rows.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 12, 12, 1, 1, 76, 347, 76, 1, 1, 701, 20429, 20429, 701, 1, 1, 8477, 1919660, 10707908, 1919660, 8477, 1, 1, 126126, 259227625, 9203978774, 9203978774, 259227625, 126126, 1, 1, 2223278, 47484618291, 12099129236936, 72078431500368
Offset: 0

Views

Author

Paul D. Hanna, Mar 08 2012

Keywords

Comments

Column 1 is A060946.
Column 2 is A209425.
Row sums equal A167007.
Antidiagonal sums equal A166894.
Central terms form A209426.

Examples

			This triangle begins:
1;
1, 1;
1, 3, 1;
1, 12, 12, 1;
1, 76, 347, 76, 1;
1, 701, 20429, 20429, 701, 1;
1, 8477, 1919660, 10707908, 1919660, 8477, 1;
1, 126126, 259227625, 9203978774, 9203978774, 259227625, 126126, 1;
1, 2223278, 47484618291, 12099129236936, 72078431500368, 12099129236936, 47484618291, 2223278, 1; ...
G.f.: A(x,y) = 1 + (1+y)*x + (1+3*y+y^2)*x^2 + (1+12*y+12*y^2+y^3)*x^3 + (1+76*y+20429*y^2+76*y^3+y^4)*x^4 +...
The logarithm of the g.f. equals the series:
log(A(x,y)) = (1 + y)*x
+ (1 + 2^2*y + y^2)*x^2/2
+ (1 + 3^3*y + 3^3*y^2 + y^3)*x^3/3
+ (1 + 4^4*y + 6^4*y^2 + 4^4*y^3 + y^4)*x^4/4
+ (1 + 5^5*y + 10^5*y^2 + 10^5*y^3 + 5^5*y^4 + y^5)*x^5/5 +...
in which the coefficients are found in triangle A209427.

Links

Crossrefs

Cf. A060946, A209425, A167007, A166894, A209426, A209427, A209196.

Programs

  • PARI
    {T(n,k)=polcoeff(polcoeff(exp(sum(m=1,n,x^m/m*sum(k=0,m,binomial(m,k)^m*y^k))+x*O(x^n)),n,x),k,y)}
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))
Showing 1-7 of 7 results.

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

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