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.

A201952 A diagonal of irregular triangle A201949.

Original entry on oeis.org

1, 1, 5, 24, 139, 945, 7377, 65016, 638418, 6910650, 81747665, 1049089470, 14516096009, 215419836359, 3412889885571, 57492203734320, 1026121982213480, 19342642266760680, 383995631680561234, 8007915240045479980, 175020604366224762038, 4000551483475536398178
Offset: 1

Views

Author

Paul D. Hanna, Dec 06 2011

Keywords

Comments

G.f. of row n in triangle A201949 equals Product_{k=0..n-1} (1 + k*x + x^2).

Examples

			E.g.f.: A(x) = x + x^2/2! + 5*x^3/3! + 24*x^4/4! + 139*x^5/5! + 945*x^6/6! + 7377*x^7/7! + 65016*x^8/8! + 638418*x^9/9! + 6910650*x^10/10! + ...
Triangle A201949 begins:
[1],
[(1), 0, 1],
[1,(1), 2, 1, 1],
[1, 3, (5), 6, 5, 3, 1],
[1, 6, 15, (24), 28, 24, 15, 6, 1],
[1, 10, 40, 90,(139), 160, 139, 90, 40, 10, 1], ...
where coefficients in parenthesis form the initial terms of this sequence.

Crossrefs

Cf. A201949, A201950, A201951, A201953.

Programs

  • PARI
    {a(n) = polcoeff( prod(j=0, n-1, 1 + j*x + x^2), n-1)}
for(n=1,30,print1(a(n),", "))

Formula

E.g.f.: Sum_{n>=0} -log(1 - x)^(2*n+1) / (n!*(n+1)!). - Paul D. Hanna, Feb 25 2019
a(n) = [x^(n-1)] Product_{k=0..n-1} (1 + k*x + x^2).
a(n) = (n-1)*a(n-1) + A201950(n-1) + A201953(n-1).

Extensions

Offset changed to 1 to agree with the e.g.f. - Paul D. Hanna, Feb 25 2019

A201953 A diagonal of irregular triangle A201949.

Original entry on oeis.org

1, 3, 15, 90, 629, 5019, 45030, 448776, 4919321, 58825415, 762089899, 10633219662, 158974192987, 2535484008225, 42970371055268, 771162539117408, 14609924404202130, 291386317037291622, 6102681801481066642, 133910606028043519500, 3072216586896101950757
Offset: 2

Views

Author

Paul D. Hanna, Dec 06 2011

Keywords

Comments

G.f. of row n in triangle A201949 equals Product_{k=0..n-1} (1 + k*x + x^2).

Examples

			E.g.f.: A(x) = x^2/2! + 3*x^3/3! + 15*x^4/4! + 90*x^5/5! + 629*x^6/6! + 5019*x^7/7! + 45030*x^8/8! + 448776*x^9/9! + 4919321*x^10/10! + ...
Triangle A201949 begins:
[1],
[1, 0, 1],
[(1), 1, 2, 1, 1],
[1,(3), 5, 6, 5, 3, 1],
[1, 6, (15), 24, 28, 24, 15, 6, 1],
[1, 10, 40, (90), 139, 160, 139, 90, 40, 10, 1],
[1, 15, 91, 300, (629), 945, 1078, 945, 629, 300, 91, 15, 1],  ...
where coefficients in parenthesis form the initial terms of this sequence.

Crossrefs

Cf. A201949, A201950, A201951, A201952.

Programs

  • PARI
    {a(n) = polcoeff( prod(j=0, n-1, 1 + j*x + x^2), n-2)}
for(n=2,30,print1(a(n),", "))

Formula

E.g.f.: Sum_{n>=0} log(1 - x)^(2*n+2) / (n!*(n+2)!). - Paul D. Hanna, Feb 25 2019
a(n) = [x^(n-2)] Product_{k=0..n-1} (1 + k*x + x^2).

Extensions

Offset changed to 2 to agree with the e.g.f. - Paul D. Hanna, Feb 25 2019

A201950 Central coefficients in Product_{k=0..n-1} (1 + k*x + x^2).

Original entry on oeis.org

1, 0, 2, 6, 28, 160, 1078, 8358, 73260, 716112, 7721844, 91039740, 1164932470, 16077368580, 238037983558, 3763371442530, 63276351409092, 1127406030014112, 21218146474666864, 420611921077524912, 8759617763834095796, 191208185756772875880
Offset: 0

Views

Author

Paul D. Hanna, Dec 06 2011

Keywords

Examples

			The coefficients in Product_{k=0..n-1} (1+k*x+x^2) form triangle A201949:
(1);
1,(0), 1;
1, 1,(2), 1, 1;
1, 3, 5, (6), 5, 3, 1;
1, 6, 15, 24, (28), 24, 15, 6, 1;
1, 10, 40, 90, 139, (160), 139, 90, 40, 10, 1;
1, 15, 91, 300, 629, 945, (1078), 945, 629, 300, 91, 15, 1;
1, 21, 182, 861, 2520, 5019, 7377, (8358), 7377, 5019, 2520, 861, 182, 21, 1;
1, 28, 330, 2156, 8729, 23520, 45030, 65016, (73260), 65016, 45030, 23520, 8729, 2156, 330, 28, 1; ...
where coefficients in parenthesis form the initial terms of this sequence.

Links

Crossrefs

Cf. A201949, A201951, A201952, A201953.
Cf. A086672, A324304 (variant).

Programs

  • Mathematica
    Flatten[{1,Table[Coefficient[Expand[Product[1 + k*x + x^2,{k,0,n-1}]],x^n],{n,1,20}]}] (* Vaclav Kotesovec, Feb 10 2015 *)
  • PARI
    {a(n) = polcoeff( prod(k=1,n,1+(k-1)*x+x^2+x*O(x^n)), n)}
for(n=0,30, print1(a(n),", "))
  • PARI
    /* From series BesselI(0, 2*log(1 - x)), after Ilya Gutkovskiy */
{a(n) = n!*polcoeff( sum(m=0,n, log(1 - x +x*O(x^n))^(2*m)/m!^2), n)}
for(n=0,30, print1(a(n),", ")) \\ Paul D. Hanna, Feb 24 2019

Formula

Central terms of rows in irregular triangle A201949.
a(n) = (n-1)*a(n-1) + 2*A201952(n-1) for n>0. [corrected by Vaclav Kotesovec, May 04 2024]
E.g.f.: BesselI(0, 2*log(1 - x)). - Ilya Gutkovskiy, Feb 22 2019
E.g.f.: Sum_{n>=0} log(1 - x)^(2*n) / n!^2. [After Ilya Gutkovskiy - Paul D. Hanna, Feb 24 2019]

A201951 G.f.: Sum_{n>=0} x^n * Product_{k=0..n-1} (1 + k*x + x^2).

Original entry on oeis.org

1, 1, 1, 3, 6, 13, 33, 85, 234, 675, 2032, 6367, 20677, 69442, 240529, 857634, 3141970, 11808611, 45464065, 179088744, 720947705, 2962994169, 12420658682, 53061133078, 230828047288, 1021809688593, 4599749893986, 21043392417004, 97784119963565, 461277854065112
Offset: 0

Views

Author

Paul D. Hanna, Dec 06 2011

Keywords

Comments

Equals the antidiagonal sums of irregular triangle A201949.

Examples

			G.f.: A(x) = 1 + x + x^2 + 3*x^3 + 6*x^4 + 13*x^5 + 33*x^6 + 85*x^7 +...
where the g.f. equals the series:
A(x) = 1 + x*(1+x^2) + x^2*(1+x^2)*(1+x+x^2) + x^3*(1+x^2)*(1+x+x^2)*(1+2*x+x^2) + x^4*(1+x^2)*(1+x+x^2)*(1+2*x+x^2)*(1+3*x+x^2) +...

Links

Crossrefs

Cf. A201949, A201950.

Programs

  • PARI
    {a(n)=sum(k=0,n,polcoeff(prod(j=0,n-k-1,1+j*x+x^2),k))}
  • PARI
    {a(n)=polcoeff(sum(m=0,n,x^m*prod(j=0,m-1,1+j*x+x^2))+x*O(x^n),n)}
  • PARI
    {a(n)=local(CF=x+x*O(x)); for(k=1, n, CF=x*(1+(n-k)*x+x^2)/(1+x*(1+(n-k)*x+x^2)-CF)); polcoeff(1/(1-CF), n, x)}

Formula

G.f.: A(x) = 1/(1 - x*(1+x^2)/(1+x*(1+x^2) - x*(1+x+x^2)/(1+x*(1+x+x^2) - x*(1+2*x+x^2)/(1+x*(1+2*x+x^2) - x*(1+3*x+x^2)/(1+x*(1+3*x+x^2) +...))))), a continued fraction.
G.f.: A(x) =1 + x*(1+x^2)/(G(0) - x*(1+x^2)) ; G(k)= k*x^2 + 1 + x + x^3 - x*(1+x+x^2+x*k)/G(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Dec 28 2011

A249790 Triangle in which row n lists the coefficients in Product_{k=1..n} (1 + k*x + x^2), for n>=0, as read by rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 4, 3, 1, 1, 6, 14, 18, 14, 6, 1, 1, 10, 39, 80, 100, 80, 39, 10, 1, 1, 15, 90, 285, 539, 660, 539, 285, 90, 15, 1, 1, 21, 181, 840, 2339, 4179, 5038, 4179, 2339, 840, 181, 21, 1, 1, 28, 329, 2128, 8400, 21392, 36630, 43624, 36630, 21392, 8400, 2128, 329, 28, 1, 1, 36, 554, 4788, 25753, 90720, 216166, 358056, 422252
Offset: 0

Views

Author

Paul D. Hanna, Nov 05 2014

Keywords

Examples

			Triangle begins:
1;
1, 1, 1;
1, 3, 4, 3, 1;
1, 6, 14, 18, 14, 6, 1;
1, 10, 39, 80, 100, 80, 39, 10, 1;
1, 15, 90, 285, 539, 660, 539, 285, 90, 15, 1;
1, 21, 181, 840, 2339, 4179, 5038, 4179, 2339, 840, 181, 21, 1;
1, 28, 329, 2128, 8400, 21392, 36630, 43624, 36630, 21392, 8400, 2128, 329, 28, 1;
1, 36, 554, 4788, 25753, 90720, 216166, 358056, 422252, 358056, 216166, 90720, 25753, 4788, 554, 36, 1;
1, 45, 879, 9810, 69399, 327285, 1058399, 2394270, 3860922, 4516380, 3860922, 2394270, 1058399, 327285, 69399, 9810, 879, 45, 1;
1, 55, 1330, 18645, 168378, 1031085, 4400648, 13305545, 28862021, 45519870, 52885644, 45519870, 28862021, 13305545, 4400648, 1031085, 168378, 18645, 1330, 55, 1; ...

Links

Crossrefs

Cf. A201826 (central coefficients), A202474 (a diagonal), A202476, A001710 (row sums).
Cf. A201949 (variant), A324956.

Programs

  • PARI
    {T(n,k)=polcoeff(prod(m=1, n, 1 + m*x + x^2 +x*O(x^k)), k,x)}
for(n=0,10,for(k=0,2*n,print1(T(n,k),", "));print(""))

Formula

E.g.f.: 1/(1 - x*y)^(1/y + 1 + y). - Paul D. Hanna, Mar 02 2019
E.g.f.: A(x,y) = 1/(1-x*y) * Sum_{k>=0} (1/y^k + y^k)/2^(0^k) * Sum_{n>=0} (-log(1 - x*y))^(2*n+k) / (n!*(n+k)!). - Paul D. Hanna, Mar 02 2019
E.g.f. of diagonal k: (1/y^k)/(1-x*y) * Sum_{n>=0} (-log(1 - x*y))^(2*n+k) / (n!*(n+k)!) for k >= 0. - Paul D. Hanna, Mar 02 2019
E.g.f.: A(x,y) = x / Series_Reversion( F(x,y) ) such that F(x/A(x,y),y) = x, where F(x,y) = Sum_{n>=1} x^n/n! * Product_{k=0..n-2} (n + (n+k)*y + n*y^2). - Paul D. Hanna, Mar 02 2019

A291845 Triangle, read by rows, where the g.f. of row n equals Product_{k=0..n-1} (1 + (2*k+1)*x + x^2) for n>0 with a single '1' in row 0.

Original entry on oeis.org

1, 1, 1, 1, 1, 4, 5, 4, 1, 1, 9, 26, 33, 26, 9, 1, 1, 16, 90, 224, 283, 224, 90, 16, 1, 1, 25, 235, 1050, 2389, 2995, 2389, 1050, 235, 25, 1, 1, 36, 511, 3660, 14174, 30324, 37723, 30324, 14174, 3660, 511, 36, 1, 1, 49, 980, 10339, 62265, 218246, 446109, 551047, 446109, 218246, 62265, 10339, 980, 49, 1, 1, 64, 1716, 25088, 218330, 1162560, 3782064, 7460928, 9157923, 7460928, 3782064, 1162560, 218330, 25088, 1716, 64, 1, 1, 81, 2805, 54324, 646542, 4899258, 23763914, 72918576, 139775763, 170606547, 139775763, 72918576, 23763914, 4899258, 646542, 54324, 2805, 81, 1
Offset: 0

Views

Author

Paul D. Hanna, Sep 03 2017

Keywords

Comments

Row sums yield the odd double factorials A001147.
Central terms in rows form A291846.
Another diagonal forms A291847.
Antidiagonal sums yield A291848.

Examples

			This irregular triangle begins:
1;
1, 1, 1;
1, 4, 5, 4, 1;
1, 9, 26, 33, 26, 9, 1;
1, 16, 90, 224, 283, 224, 90, 16, 1;
1, 25, 235, 1050, 2389, 2995, 2389, 1050, 235, 25, 1;
1, 36, 511, 3660, 14174, 30324, 37723, 30324, 14174, 3660, 511, 36, 1;
1, 49, 980, 10339, 62265, 218246, 446109, 551047, 446109, 218246, 62265, 10339, 980, 49, 1;
1, 64, 1716, 25088, 218330, 1162560, 3782064, 7460928, 9157923, 7460928, 3782064, 1162560, 218330, 25088, 1716, 64, 1;
1, 81, 2805, 54324, 646542, 4899258, 23763914, 72918576, 139775763, 170606547, 139775763, 72918576, 23763914, 4899258, 646542, 54324, 2805, 81, 1;
1, 100, 4345, 107700, 1681503, 17237880, 117496358, 529332200, 1548992621, 2899264620, 3521075919, 2899264620, 1548992621, 529332200, 117496358, 17237880, 1681503, 107700, 4345, 100, 1; ...

Links

Crossrefs

Cf. A291846, A291847, A291848, A201949, A001147 (row sums).

Programs

  • PARI
    {T(n, k)=polcoeff(prod(j=0, n-1, 1 + (2*j+1)*x + x^2), k)}
{for(n=0, 10, for(k=0, 2*n, print1(T(n, k), ", ")); print(""))}

A324305 Triangle, read by rows, where the g.f. of row n equals Product_{k=0..n-2} (n + k*y + n*y^2) for n > 1 with a single '1' in row 1.

Original entry on oeis.org

1, 2, 0, 2, 9, 3, 18, 3, 9, 64, 48, 200, 96, 200, 48, 64, 625, 750, 2775, 2280, 4300, 2280, 2775, 750, 625, 7776, 12960, 46440, 53640, 100584, 81360, 100584, 53640, 46440, 12960, 7776, 117649, 252105, 909979, 1337700, 2594501, 2753415, 3604342, 2753415, 2594501, 1337700, 909979, 252105, 117649, 2097152, 5505024, 20414464, 36040704, 73543680, 94730496, 133244544, 128389632, 133244544, 94730496, 73543680, 36040704, 20414464, 5505024, 2097152
Offset: 1

Views

Author

Paul D. Hanna, Feb 28 2019

Keywords

Examples

			E.g.f.: A(x,y) = Sum_{n>=1} x^n/n! * Product_{k=0..n-2} (n + k*y + n*y^2) and satisfies A(x,y) = x/(1 - y*A(x,y))^(1/y + y).
Explicitly,
A(x,y) = x + (2*y^2 + 2)*x^2/2! + (9*y^4 + 3*y^3 + 18*y^2 + 3*y + 9)*x^3/3! + (64*y^6 + 48*y^5 + 200*y^4 + 96*y^3 + 200*y^2 + 48*y + 64)*x^4/4! + (625*y^8 + 750*y^7 + 2775*y^6 + 2280*y^5 + 4300*y^4 + 2280*y^3 + 2775*y^2 + 750*y + 625)*x^5/5! + (7776*y^10 + 12960*y^9 + 46440*y^8 + 53640*y^7 + 100584*y^6 + 81360*y^5 + 100584*y^4 + 53640*y^3 + 46440*y^2 + 12960*y + 7776)*x^6/6! + (117649*y^12 + 252105*y^11 + 909979*y^10 + 1337700*y^9 + 2594501*y^8 + 2753415*y^7 + 3604342*y^6 + 2753415*y^5 + 2594501*y^4 + 1337700*y^3 + 909979*y^2 + 252105*y + 117649)*x^7/7! + ...
Setting y = 1 yields an o.g.f. of A006013:
A(x,y=1) = x + 2*x^2 + 7*x^3 + 30*x^4 + 143*x^5 + 728*x^6 + 3876*x^7 + 21318*x^8 + 120175*x^9 + ... + binomial(3*n-2,n-1)/n * x^n + ...
TRIANGLE.
This triangle of coefficients in Product_{k=0..n-2} (n + k*y + n*y^2), n >= 1, begins
1;
2, 0, 2;
9, 3, 18, 3, 9;
64, 48, 200, 96, 200, 48, 64;
625, 750, 2775, 2280, 4300, 2280, 2775, 750, 625;
7776, 12960, 46440, 53640, 100584, 81360, 100584, 53640, 46440, 12960, 7776;
117649, 252105, 909979, 1337700, 2594501, 2753415, 3604342, 2753415, 2594501, 1337700, 909979, 252105, 117649;
2097152, 5505024, 20414464, 36040704, 73543680, 94730496, 133244544, 128389632, 133244544, 94730496, 73543680, 36040704, 20414464, 5505024, 2097152; ...
RELATED SERIES.
The e.g.f. may be defined by A(x,y) = Series_Reversion( x/G(x,y) )
where G(x,y) is the e.g.f. of A201949 and equals
G(x,y) = Sum_{n>=0} x^n/n! * Product_{k=0..n-1} (1 + k*y + y^2)
so that
G(x,y) = 1 + (1 + y^2)*x + (1 + y + 2*y^2 + y^3 + y^4)*x^2/2! + (1 + 3*y + 5*y^2 + 6*y^3 + 5*y^4 + 3*y^5 + y^6)*x^3/3! + (1 + 6*y + 15*y^2 + 24*y^3 + 28*y^4 + 24*y^5 + 15*y^6 + 6*y^7 + y^8)*x^4/4! + (1 + 10*y + 40*y^2 + 90*y^3 + 139*y^4 + 160*y^5 + 139*y^6 + 90*y^7 + 40*y^8 + 10*y^9 + y^10)*x^5/5! + ...
and G(x,y) = x / Series_Reversion( A(x,y) ).
RELATED TRIANGLE.
Triangle A201949 of coefficients in G(x,y) such that A(x/G(x,y),y) = x begins
1;
1, 0, 1;
1, 1, 2, 1, 1;
1, 3, 5, 6, 5, 3, 1;
1, 6, 15, 24, 28, 24, 15, 6, 1;
1, 10, 40, 90, 139, 160, 139, 90, 40, 10, 1;
1, 15, 91, 300, 629, 945, 1078, 945, 629, 300, 91, 15, 1; ...
where the g.f. of row n is Product_{k=0..n-1} (1 + k*y + y^2) for n >= 0.

Links

Crossrefs

Cf. A201949, A324304.
Cf. A324956, A324958.

Programs

  • PARI
    {T(n, k)=polcoeff(prod(j=0, n-2,  n + j*y + n*y^2), k, y)}
{for(n=1, 10, for(k=0, 2*n-2, print1(T(n, k), ", ")); print(""))}
  • PARI
    /* A(x,y) = Series_Reversion(x/G(x,y)) where G(x,y) = e.g.f. A201949 */
{T(n,k) = my(G=1,A=x);
G = sum(m=0,n, x^m/m! * prod(j=0,m-1, 1 + j*y + y^2) +x*O(x^n));
A = serreverse(x/G);
n!*polcoeff(polcoeff(A,n,x),k,y)}
{for(n=1, 10, for(k=0, 2*n-2, print1(T(n, k), ", ")); print(""))}

Formula

GENERATING FUNCTIONS.
E.g.f.: A(x,y) = x/(1 - y*A(x,y))^(1/y + y).
E.g.f.: A(x,y) = Series_Reversion( x*(1 - x*y)^(1/y + y) ), wrt x.
E.g.f.: A(x,y) = Sum_{n>=1} x^n/n! * Product_{k=0..n-2} (n + k*y + n*y^2)
E.g.f.: A(x,y) = Series_Reversion( x/G(x,y) ) such that A(x/G(x,y),y) = x, where G(x,y) = Sum_{n>=0} x^n/n! * Product_{k=0..n-1} (1 + k*y + y^2) is the e.g.f. of A201949.
PARTICULAR ARGUMENTS.
E.g.f. at y = 0: A(x,y=0) = -LambertW(-x) = x*exp(-LambertW(-x)).
E.g.f. at y = 1: A(x,y=1) = x*G(x)^2, where G = 1 + x*G(x)^3 is the g.f. of A001764.
FORMULAS INVOLVING TERMS.
Row sums: Sum_{k=0..2*n-2} T(n,k) = (3*n-2)!/(2*n-1)! for n >= 1.
T(n,0) = T(n,2*n-2) = n^(n-1) for n >= 1.
T(n,n-1) = A324304(n) for n >= 1.
Showing 1-7 of 7 results.

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