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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.

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A220265 Triangle where the g.f. of row n is: Sum_{k=0..n^2-n+1} T(n,k)*y^k = (2*(1+y)^n - 1) * ((1+y)^n - 1)^(n-1) / y^(n-1), as read by rows.

Original entry on oeis.org

1, 2, 2, 9, 8, 2, 9, 72, 177, 222, 163, 72, 18, 2, 64, 800, 3696, 9800, 17408, 22284, 21340, 15554, 8652, 3633, 1120, 240, 32, 2, 625, 11250, 82500, 365000, 1131750, 2654250, 4922750, 7425000, 9274150, 9704600, 8566200, 6398000, 4042345, 2152890, 959690, 354020
Offset: 1

Views

Author

Paul D. Hanna, Dec 09 2012

Keywords

Comments

Based on the identity:
1 = Sum_{n>=1} (2*G(x)^n - 1) * (1 - G(x)^n)^(n-1) for all G(x) such that G(0)=1.

Examples

			Triangle begins:
1, 2;
2, 9, 8, 2;
9, 72, 177, 222, 163, 72, 18, 2;
64, 800, 3696, 9800, 17408, 22284, 21340, 15554, 8652, 3633, 1120, 240, 32, 2;
625, 11250, 82500, 365000, 1131750, 2654250, 4922750, 7425000, 9274150, 9704600, 8566200, 6398000, 4042345, 2152890, 959690, 354020, 106251, 25300, 4600, 600, 50, 2;
7776, 190512, 2015280, 13222440, 62141310, 225598527, 662159412, 1618976925, 3366367410, 6041884575, 9462175520, 13034476980, 15886286910, 17202209995, 16595155500, 14285514705, 10978477070, 7528219125, 4599186000, 2496823900, 1200043026, 508072257, 188241900, 60515895, 16695030, 3895573, 753984, 117810, 14280, 1260, 72, 2; ...
where the alternating antidiagonal sums equal zero (after the initial '1'):
0 = 2 - 2;
0 = 9 - 9;
0 = 64 - 72 + 8;
0 = 625 - 800 + 177 - 2;
0 = 7776 - 11250 + 3696 - 222;
0 = 117649 - 190512 + 82500 - 9800 + 163; ...
Column 0 forms A000169(n) = n^(n-1) and column 1 equals n^(n-2)*n*(n+1)^2/2.
The largest term in row n, found at position ceiling(n^2/2) - (n-1), begins:
[2, 9, 222, 22284, 9704600, 17202209995, 123106610062800, 3600033286934164416, 421003580776636784633028, 200645860378226792820279591852, ...].

Links

Crossrefs

Cf. A220266, A220267, A000169.

Programs

  • PARI
    {T(n,k)=polcoeff((2*(1+x)^n-1)*((1+x)^n-1)^(n-1)/x^(n-1),k)}
for(n=1,6,for(k=0,n^2-n+1,print1(T(n,k),", "));print(("")))

Formula

0 = Sum_{k=0..n-1} (-1)^k * T(n-k,k) for n>1.
Antidiagonal sums equal A220266.
Main diagonal equals A220267.
Row sums equal (2^(n+1) - 1)*(2^n - 1)^(n-1).
Position of largest term in row n is: A099392(n) = ceiling(n^2/2) - (n-1).

A220231 G.f.: Sum_{n>=1} (2 - (1-x)^n) * (1 - (1-x)^n)^(n-1) / (1-x)^(n^2).

Original entry on oeis.org

1, 4, 22, 184, 2094, 30300, 532918, 11041936, 263451958, 7114155240, 214501732236, 7143254501280, 260397686968140, 10313589262630724, 441025376697310226, 20250545724670230240, 993756093394720170206, 51903607891071021948092, 2874779134773855301743682
Offset: 0

Views

Author

Paul D. Hanna, Dec 11 2012

Keywords

Comments

Compare the g.f. of this sequence to the identity (when G(x) = 1/(1-x)):
1 = Sum_{n>=1} (2*G(x)^n - 1) * (1 - G(x)^n)^(n-1) for all G(x) such that G(0)=1.

Examples

			G.f.: A(x) = 1 + 4*x + 22*x^2 + 184*x^3 + 2094*x^4 + 30300*x^5 +...
where
A(x) = (1+x)/(1-x) + (1+2*x-x^2)*(2*x-x^2)/(1-x)^4 + (1+3*x-3*x^2+x^3)*(3*x-3*x^2+x^3)^2/(1-x)^9 + (1+4*x-6*x^2+4*x^3-x^4)*(4*x-6*x^2+4*x^3-x^4)^3/(1-x)^16 +...
Compare the g.f. to the identity:
1 = (1+x)/(1-x) - (1+2*x-x^2)*(2*x-x^2)/(1-x)^4 + (1+3*x-3*x^2+x^3)*(3*x-3*x^2+x^3)^2/(1-x)^9 - (1+4*x-6*x^2+4*x^3-x^4)*(4*x-6*x^2+4*x^3-x^4)^3/(1-x)^16 +-...

Crossrefs

Cf. A220266.

Programs

  • PARI
    {a(n)=polcoeff(sum(m=1, n+1, (2 - (1-x)^m) * (1 - (1-x)^m +x*O(x^n))^(m-1)/(1-x+x*O(x^n))^(m^2)), n)}
for(n=0, 20, print1(a(n), ", "))
  • PARI
    {a(n)=polcoeff(1+sum(m=1, n\2+1, 2*(2 - (1-x)^(2*m)) * (1 - (1-x)^(2*m) +x*O(x^n))^(2*m-1)/(1-x+x*O(x^n))^(4*m^2)), n)}
for(n=0, 20, print1(a(n), ", "))
  • PARI
    {a(n)=polcoeff(-1+sum(m=0, n\2, 2*(2 - (1-x)^(2*m+1)) * (1 - (1-x)^(2*m+1) +x*O(x^n))^(2*m)/(1-x+x*O(x^n))^((2*m+1)^2)), n)}
for(n=0, 20, print1(a(n), ", "))

Formula

G.f.: 1 + Sum_{n>=1} 2*(2 - (1-x)^(2*n)) * (1 - (1-x)^(2*n))^(2*n-1) / (1-x)^(4*n^2).
G.f.: -1 + Sum_{n>=0} 2*(2 - (1-x)^(2*n+1))*(1 - (1-x)^(2*n+1))^(2*n)/(1-x)^((2*n+1)^2).

A220267 Main diagonal of triangle A220265.

Original entry on oeis.org

1, 9, 177, 9800, 1131750, 225598527, 69153712446, 30211650109440, 17832410391617082, 13670065258130703125, 13203133188522251881137, 15685246720965582029887488, 22477297594725738707224270105, 38231902029181930196176183861755, 76144787589417130318451646093750000
Offset: 1

Views

Author

Paul D. Hanna, Dec 09 2012

Keywords

Comments

G.f. of row n of triangle A220265 equals: Sum_{k=0..n^2-n+1} A220265(n,k)*y^k = (2*(1+y)^n - 1)*((1+y)^n - 1)^(n-1)/y^(n-1) for n>=1.

Examples

			 Irregular triangle A220265 begins:
1, 2;
2, 9, 8, 2;
9, 72, 177, 222, 163, 72, 18, 2;
64, 800, 3696, 9800, 17408, 22284, 21340, 15554, 8652, 3633, 1120, 240, 32, 2;
625, 11250, 82500, 365000, 1131750, 2654250, 4922750, 7425000, 9274150, 9704600, 8566200, 6398000, 4042345, 2152890, 959690, 354020, 106251, 25300, 4600, 600, 50, 2; ...

Crossrefs

Cf. A220265, A220266.

Programs

  • PARI
    {a(n)=polcoeff((2*(1+x)^n-1)*((1+x)^n-1)^(n-1)/x^(n-1), n-1)}
for(n=1, 20, print1(a(n), ", "))

Formula

a(n) = A220265(n,n-1) for n>=1.
Showing 1-3 of 3 results.

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