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.

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]

A201949 Triangle, read by rows, where the g.f. of row n equals Product_{k=0..n-1} (1 + k*y + y^2) for n>0 with a single '1' in row 0.

Original entry on oeis.org

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
Offset: 0

Views

Author

Paul D. Hanna, Dec 06 2011

Keywords

Comments

The formula for the main diagonal, BesselI(0, 2*log(1 - x)), was found by Ilya Gutkovskiy (see A201950). - Paul D. Hanna, Feb 24 2019

Examples

			E.g.f.: A(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! + ...
which equals the power series expansion in x of the series given by
A(x,y)  =  Sum_{n>=0} log(1 - x*y)^(2*n) / (n!^2)  -  (1/y + y) * Sum_{n>=0} log(1 - x*y)^(2*n+1) / (n!*(n+1)!)  +  (1/y^2 + y^2) * Sum_{n>=0} log(1 - x*y)^(2*n+2) / (n!*(n+2)!)  -  (1/y3 + y^3) * Sum_{n>=0} (-log(1 - x*y))^(2*n+3) / (n!*(n+3)!)  +  (1/y^4 + y^4) * Sum_{n>=0} log(1 - x*y)^(2*n+4) / (n!*(n+4)!) + ...
Triangle 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],
[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], ...
such that the g.f. of row n equals Product_{k=0..n-1} (1 + k*x + x^2) for n>0.
RELATED SERIES.
The e.g.f. may be defined by A(x,y) = x / Series_Reversion( F(x,y) )
where F(x,y) is the e.g.f. of triangle A324305 and equals
F(x,y) = Sum_{n>=1} x^n/n! * Product_{k=0..n-2} (n + k*y + n*y^2)
so that
F(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! + ...
where F(x,y) = Series_Reversion( x/A(x,y) ).
RELATED TRIANGLE.
Triangle A324305 of coefficients in F(x,y) such that F(x/A(x,y),y) = x 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; ...
where the g.f. of row n is Product_{k=0..n-2} (n + k*y + n*y^2) for n >= 1.

Links

Crossrefs

Cf. A201950, A201951; diagonals: A201952, A201953.
Cf. A324305.

Programs

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

Formula

Row sums yield the factorials.
Central terms in rows form A201950.
Antidiagonal sums yield A201951.
GENERATING FUNCTIONS.
E.g.f.: A(x,y) = 1/(1 - x*y)^(1/y + y). - Paul D. Hanna, Mar 02 2019
E.g.f.: A(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, Feb 24 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 + k*y + n*y^2) is the e.g.f. of A324305. - Paul D. Hanna, Feb 28 2019
E.g.f. of diagonal k: (1/y^k) * Sum_{n>=0} (-log(1 - x*y))^(2*n+k) / (n!*(n+k)!) for k >= 0. - Paul D. Hanna, Feb 24 2019
PARTICULAR ARGUMENTS.
E.g.f. at y = 0: A(x,y=0) = exp(x).
E.g.f. at y = 1: A(x,y=1) = 1/(1-x)^2.
E.g.f. at y = 2: A(x,y=2) = 1/(1-2*x)^(5/2).

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

A202476 G.f.: Sum_{n>=0} x^n * Product_{k=1..n} (1 + k*x + x^2).

Original entry on oeis.org

1, 1, 2, 5, 11, 28, 74, 206, 601, 1826, 5766, 18851, 63676, 221678, 793958, 2920292, 11014653, 42543773, 168074091, 678403932, 2794920078, 11742254750, 50266213000, 219085792538, 971543475593, 4380664101448, 20071848941411, 93403455862117, 441206005123701
Offset: 0

Views

Author

Paul D. Hanna, Dec 19 2011

Keywords

Examples

			The coefficients in Product_{k=1..n} (1+k*x+x^2), n>=0, form the triangle:
[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]; ...
the antidiagonal sums of which form this sequence.

Links

Crossrefs

Cf. A201951, A201826, A202474.

Programs

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

Formula

Antidiagonal sums of the irregular triangle in which row n is defined by the g.f.: Product_{k=1..n} (1 + k*x + x^2) for n>=0.
G.f.: 1/(1 - 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) - x*(1+4*x+x^2)/(1 + x*(1+4*x+x^2) -...))))), a continued fraction.

A291848 G.f.: Sum_{n>=0} x^n * Product_{k=0..n-1} (1 + (2*k+1)*x + x^2).

Original entry on oeis.org

1, 1, 2, 6, 15, 47, 150, 522, 1903, 7319, 29396, 122988, 534141, 2400061, 11136516, 53220492, 261576725, 1319629445, 6825232486, 36137198722, 195664517227, 1082169511883, 6108213101658, 35153836421302, 206126910439763, 1230477025952427, 7473067121404104, 46146114390128888, 289554642297817561, 1845220293901278041, 11936266843924805064
Offset: 0

Views

Author

Paul D. Hanna, Sep 03 2017

Keywords

Comments

Antidiagonal sums of irregular triangle A291845, which has row sums equal to the odd double factorials A001147.

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 15*x^4 + 47*x^5 + 150*x^6 + 522*x^7 + 1903*x^8 + 7319*x^9 + 29396*x^10 + 122988*x^11 + 534141*x^12 +...
which equals the series:
A(x) = 1 + x*(1+x+x^2) + x^2*(1+x+x^2)*(1+3*x+x^2) + x^3*(1+x+x^2)*(1+3*x+x^2)*(1+5*x+x^2) + x^4*(1+x+x^2)*(1+3*x+x^2)*(1+5*x+x^2)*(1+7*x+x^2) +...

Links

Crossrefs

Cf. A291845, A201951.

Programs

  • Mathematica
    nmax = 30; CoefficientList[Series[Sum[2^n*x^(2*n)*Pochhammer[(1 + x + x^2)/(2*x), n], {n, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 13 2017 *)
  • PARI
    {a(n)=sum(k=0, n, polcoeff(prod(j=0, n-k-1, 1+(2*j+1)*x+x^2), k))}
for(n=0,30,print1(a(n),", "))
  • PARI
    {a(n)=polcoeff(sum(m=0, n, x^m*prod(j=0, m-1, 1+(2*j+1)*x+x^2))+x*O(x^n), n)}
for(n=0,25,print1(a(n),", "))

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
Showing 1-6 of 6 results.

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