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.

A228906 A diagonal of triangle A228904.

Original entry on oeis.org

1, 2, 7, 62, 1031, 24782, 774180, 29763855, 1359654560, 71984907423, 4335406418694, 292753300447894, 21909289407621069, 1800106653483146619, 161097567109713138999, 15599377878403186676330, 1625083531855929644443019, 181238001661004834528467994, 21545324993880123460418461719
Offset: 1

Views

Author

Paul D. Hanna, Sep 07 2013

Keywords

Comments

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

Crossrefs

Cf. A228904.

Programs

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

A228832 Triangle defined by T(n,k) = binomial(n*k, k^2), for n>=0, k=0..n, as read by rows.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 15, 1, 1, 4, 70, 220, 1, 1, 5, 210, 5005, 4845, 1, 1, 6, 495, 48620, 735471, 142506, 1, 1, 7, 1001, 293930, 30421755, 183579396, 5245786, 1, 1, 8, 1820, 1307504, 601080390, 40225345056, 69668534468, 231917400, 1, 1, 9, 3060, 4686825, 7307872110, 3169870830126, 96926348578605, 37387265592825, 11969016345, 1
Offset: 0

Views

Author

Paul D. Hanna, Sep 04 2013

Keywords

Comments

Central coefficients are A201555(n) = C(2*n^2,n^2) = A000984(n^2), where A000984 is the central binomial coefficients.

Examples

			The triangle of coefficients C(n*k, k^2), n>=k, k=0..n, begins:
1;
1, 1;
1, 2, 1;
1, 3, 15, 1;
1, 4, 70, 220, 1;
1, 5, 210, 5005, 4845, 1;
1, 6, 495, 48620, 735471, 142506, 1;
1, 7, 1001, 293930, 30421755, 183579396, 5245786, 1;
1, 8, 1820, 1307504, 601080390, 40225345056, 69668534468, 231917400, 1;
1, 9, 3060, 4686825, 7307872110, 3169870830126, 96926348578605, 37387265592825, 11969016345, 1; ...

Links

Crossrefs

Cf. A228808 (row sums), A228833 (antidiagonal sums), A135860 (diagonal), A201555 (central terms).
Cf. A229052.
Cf. related triangles: A228904 (exp), A209330, A226234, A228836.

Programs

  • PARI
    {T(n, k)=binomial(n*k, k^2)}
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(""))

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), ", "))

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), ", "))

A228899 Triangle defined by g.f. A(x,y) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n, k)^(k+1) * y^k ), as read by rows.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 6, 12, 1, 1, 10, 71, 76, 1, 1, 15, 281, 2153, 701, 1, 1, 21, 861, 29166, 129509, 8477, 1, 1, 28, 2212, 244725, 7664343, 12391414, 126126, 1, 1, 36, 4998, 1477391, 218030412, 3875325345, 1699148352, 2223278, 1, 1, 45, 10242, 7017577, 3748460115, 448713017405, 3284369541969, 315158247170, 45269999, 1
Offset: 0

Views

Author

Paul D. Hanna, Sep 07 2013

Keywords

Comments

Note that the following g.f. does NOT yield an integer triangle: exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n, k)^k * y^k ).

Examples

			This triangle begins:
1;
1, 1;
1, 3, 1;
1, 6, 12, 1;
1, 10, 71, 76, 1;
1, 15, 281, 2153, 701, 1;
1, 21, 861, 29166, 129509, 8477, 1;
1, 28, 2212, 244725, 7664343, 12391414, 126126, 1;
1, 36, 4998, 1477391, 218030412, 3875325345, 1699148352, 2223278, 1;
1, 45, 10242, 7017577, 3748460115, 448713017405, 3284369541969, 315158247170, 45269999, 1; ...
...
G.f.: A(x,y) = 1 + (1+y)*x + (1+3*y+y^2)*x^2 + (1+6*y+12*y^2+y^3)*x^3 + (1+10*y+71*y^2+76*y^3+y^4)*x^4 + (1+15*y+281*y^2+2153*y^3+701*y^4+y^5)*x^5 +...
The logarithm of the g.f. equals the series:
log(A(x)) = (1 + x)*x
+ (1 + 2^2*x + x^2)*x^2/2
+ (1+ 3^2*y + 3^3*y^2 + y^3)*x^3/3
+ (1+ 4^2*y + 6^3*y^2 + 4^4*y^3 + x^4)*x^4/4
+ (1+ 5^2*y + 10^3*y^2 + 10^4*y^3 + 5^5*y^4 + y^5)*x^5/5
+ (1+ 6^2*y + 15^3*y^2 + 20^4*y^3 + 15^5*y^4 + 6^6*y^5 + y^6)*x^6/6 +...
in which the coefficients form A219207(n,k) = binomial(n, k)^(k+1).

Crossrefs

Cf. A184730 (row sums), A181070 (antidiagonal sums), A060946 (diagonal).
Cf. related triangles: A219207, A209424, A228904.

Programs

  • PARI
    {T(n, k)=polcoeff(polcoeff(exp(sum(m=1, n, x^m/m*sum(j=0, m, binomial(m, j)^(j+1)*y^j))+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.

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