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 1-3 of 3 results.

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).

A324304 a(n) = [y^(n-1)] Product_{k=0..n-2} (n + k*y + n*y^2) for n > 1 with a(1) = 1.

Original entry on oeis.org

1, 0, 18, 96, 4300, 81360, 3604342, 128389632, 6704335980, 346778956800, 21896347260084, 1459386186255360, 110117675704707190, 8898156449299703040, 786739773441598071750, 74406732202318884372480, 7565016269351818379826372, 818338704493281924572946432, 94154670956813022045927404464, 11458715042302170139584184320000, 1472412964588453156024745207931636
Offset: 1

Views

Author

Paul D. Hanna, Feb 28 2019

Keywords

Examples

			E.g.f.: A(x) = x + 18*x^3/3! + 96*x^4/4! + 4300*x^5/5! + 81360*x^6/6! + 3604342*x^7/7! + 128389632*x^8/8! + 6704335980*x^9/9! + 346778956800*x^10/10! + 21896347260084*x^11/11! + 1459386186255360*x^12/12! + ...
RELATED TRIANGLE.
Triangle A324305 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; ...
in which the central terms, A324305(n, n-1) for n >= 1, form this sequence.

Links

Crossrefs

Cf. A324305.
Cf. A201950 (variant).

Programs

  • Mathematica
    Flatten[{1, Table[Coefficient[Expand[Product[(n + k*y + n*y^2), {k, 0, n-2}]], y^(n-1)], {n, 2, 20}]}] (* Vaclav Kotesovec, Mar 13 2019 *)
  • PARI
    {A324305(n, k) = polcoeff( prod(j=0, n-2,  n + j*y + n*y^2), k, y)}
{a(n) = A324305(n, n-1)}
for(n=1, 25, print1(a(n), ", "))

Formula

a(n) = A324305(n, n-1) for n >= 1.
a(n) ~ c * n! * (27/4)^n / n^2, where c = 1/(6*Pi*sqrt(3*log(3/2))) = 0.04810181967270783985882272373499905248047631331... - Vaclav Kotesovec, Mar 13 2019, updated Mar 17 2024

A324958 Triangle of coefficients T(n,k) of y^n in Product_{k=0..n-2} (n + (2*n + k)*y + n*y^2), as read by rows of terms k = 0..2*n-2, for n >= 1.

Original entry on oeis.org

1, 2, 4, 2, 9, 39, 60, 39, 9, 64, 432, 1160, 1584, 1160, 432, 64, 625, 5750, 22275, 47380, 60460, 47380, 22275, 5750, 625, 7776, 90720, 461160, 1343160, 2479464, 3029040, 2479464, 1343160, 461160, 90720, 7776, 117649, 1663893, 10489969, 38937360, 94679711, 158760987, 188149822, 158760987, 94679711, 38937360, 10489969, 1663893, 117649, 2097152, 34865152, 262635520, 1187049472, 3593318400, 7701010688, 12043471488, 13957194496, 12043471488, 7701010688, 3593318400, 1187049472, 262635520, 34865152, 2097152
Offset: 1

Views

Author

Paul D. Hanna, Mar 20 2019

Keywords

Examples

			E.g.f.: A(x,y) = Sum_{n>=0} x^n/n! * Sum_{k=0..2*n-2} T(n,k)*y^k starts
A(x,y) = x + (2*y^2 + 4*y + 2)*x^2/2! + (9*y^4 + 39*y^3 + 60*y^2 + 39*y + 9)*x^3/3! + (64*y^6 + 432*y^5 + 1160*y^4 + 1584*y^3 + 1160*y^2 + 432*y + 64)*x^4/4! + (625*y^8 + 5750*y^7 + 22275*y^6 + 47380*y^5 + 60460*y^4 + 47380*y^3 + 22275*y^2 + 5750*y + 625)*x^5/5! + (7776*y^10 + 90720*y^9 + 461160*y^8 + 1343160*y^7 + 2479464*y^6 + 3029040*y^5 + 2479464*y^4 + 1343160*y^3 + 461160*y^2 + 90720*y + 7776)*x^6/6! + (117649*y^12 + 1663893*y^11 + 10489969*y^10 + 38937360*y^9 + 94679711*y^8 + 158760987*y^7 + 188149822*y^6 + 158760987*y^5 + 94679711*y^4 + 38937360*y^3 + 10489969*y^2 + 1663893*y + 117649)*x^7/7! + (2097152*y^14 + 34865152*y^13 + 262635520*y^12 + 1187049472*y^11 + 3593318400*y^10 + 7701010688*y^9 + 12043471488*y^8 + 13957194496*y^7 + 12043471488*y^6 + 7701010688*y^5 + 3593318400*y^4 + 1187049472*y^3 + 262635520*y^2 + 34865152*y + 2097152)*x^8/8! + ...
This triangle of coefficients T(n,k) of x^n*y^k/n! in e.g.f. A(x,y) begins:
1;
2, 4, 2;
9, 39, 60, 39, 9;
64, 432, 1160, 1584, 1160, 432, 64;
625, 5750, 22275, 47380, 60460, 47380, 22275, 5750, 625;
7776, 90720, 461160, 1343160, 2479464, 3029040, 2479464, 1343160, 461160, 90720, 7776;
117649, 1663893, 10489969, 38937360, 94679711, 158760987, 188149822, 158760987, 94679711, 38937360, 10489969, 1663893, 117649;
2097152, 34865152, 262635520, 1187049472, 3593318400, 7701010688, 12043471488, 13957194496, 12043471488, 7701010688, 3593318400, 1187049472, 262635520, 34865152, 2097152; ...

Crossrefs

Cf. A324959, A002294.
Cf. A324960, A324305.

Programs

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

Formula

E.g.f. A(x) = Sum_{n>=1} x^n/n! * Sum_{k=0..2*n-2} T(n,k)*y^k satisfies
(1) A(x,y) = Sum_{n>=1} x^n/n! * Product_{k=0..n-2} (n + (2*n + k)*y + n*y^2).
(2) A(x,y) = Series_Reversion( x*(1 - x*y)^((1+y)^2/y) ), wrt x.
(3) A(x,y) = x/(1 - y*A(x))^((1+y)^2/y).
(4) A(x,y) = x*Sum_{n>=0} A(x,y)^n/n! * Product_{k=0..n-1} (1 + (k+2)*y + y^2).
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)^4, where G(x) = 1 + x*G(x)^5 is the g.f. of A002294.
FORMULAS INVOLVING TERMS.
Row sums: Sum_{k=0..2*n-2} T(n,k) = (5*n-2)!/(4*n-1)! for n >= 1.
T(n,0) = T(n,2*n-2) = n^(n-1) for n >= 1.
T(n,n-1) = A324959(n) for n >= 1.
Showing 1-3 of 3 results.

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