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.

A228903 Second diagonal of triangle A228902.

Original entry on oeis.org

1, 3, 45, 2905, 411500, 100545716, 37614371968, 19977489354808, 14291976911388733, 13248449270832615473, 15445843697030022753539, 22118947642938636936616739, 38166279965353127309979185861, 78098472647949324804924786089941, 186993874842690500015923551395259661
Offset: 1

Views

Author

Paul D. Hanna, Sep 07 2013

Keywords

Comments

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

Crossrefs

Cf. A228902.

Programs

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

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),", "))

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(""))

A206848 G.f.: exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n^2, k^2) ).

Original entry on oeis.org

1, 2, 5, 53, 3422, 826606, 1335470713, 9548109569885, 190076214495558260, 18558289189760778318731, 10286810587274357297985552184, 16301371794177939084545371104827679, 91249944361047494534207504939405352235731, 3283593155431496336538359592977826684908598341441
Offset: 0

Views

Author

Paul D. Hanna, Feb 15 2012

Keywords

Comments

Logarithmic derivative yields A206849.
Equals row sums of triangle A228902.

Examples

			G.f.: A(x) = 1 + 2*x + 5*x^2 + 53*x^3 + 3422*x^4 + 826606*x^5 + 1335470713*x^6 +...
where the logarithm of the g.f. yields the l.g.f. of A206849:
log(A(x)) = 2*x + 6*x^2/2 + 137*x^3/3 + 13278*x^4/4 + 4098627*x^5/5 +...

Links

Crossrefs

Cf. A206849 (log), A206846, A206850, A228902.
Cf. variants: A167006, A228809.

Programs

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

A226234 Triangle defined by T(n,k) = binomial(n^2, k^2), for n>=0, k=0..n, as read by rows.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 9, 126, 1, 1, 16, 1820, 11440, 1, 1, 25, 12650, 2042975, 2042975, 1, 1, 36, 58905, 94143280, 7307872110, 600805296, 1, 1, 49, 211876, 2054455634, 3348108992991, 63205303218876, 262596783764, 1, 1, 64, 635376, 27540584512, 488526937079580, 401038568751465792, 1118770292985239888, 159518999862720, 1
Offset: 0

Views

Author

Paul D. Hanna, Aug 24 2013

Keywords

Comments

Row sums equal A206849.
Antidiagonal sums equal A123165.

Examples

			The triangle of coefficients C(n^2,k^2), n>=k, k=0..n, begins:
1;
1, 1;
1, 4, 1;
1, 9, 126, 1;
1, 16, 1820, 11440, 1;
1, 25, 12650, 2042975, 2042975, 1;
1, 36, 58905, 94143280, 7307872110, 600805296, 1;
1, 49, 211876, 2054455634, 3348108992991, 63205303218876, 262596783764, 1;
1, 64, 635376, 27540584512, 488526937079580, 401038568751465792, 1118770292985239888, 159518999862720, 1; ...

Links

Crossrefs

Cf. A206849, A123165.
Cf. related triangles: A228902(exp), A209330, A228832, A228836.

Programs

  • PARI
    {T(n,k)=binomial(n^2,k^2)}
for(n=0,9,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(""))
Showing 1-6 of 6 results.

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

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