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 1-5 of 5 results.

A221370 O.g.f.: Sum_{n>=0} n! * x^n * Product_{k=1..n} (k + x) / (1 + k^2*x + k*x^2).

Original entry on oeis.org

1, 1, 4, 21, 183, 2362, 42449, 1012897, 30961412, 1179154241, 54727128731, 3040047461530, 199109235070645, 15182265283487213, 1333242114217704924, 133577535961042535669, 15144191953510005439455, 1928873660857769308675146, 274228718414760130917382185
Offset: 0

Views

Author

Paul D. Hanna, Jan 13 2013

Keywords

Comments

Compare to the identity:
Sum_{n>=0} n! * x^n * Product_{k=1..n} (1 + x) / (1 + k*x + k*x^2) = 1/(1-x-x^2).

Examples

			O.g.f.: A(x) = 1 + x + 4*x^2 + 21*x^3 + 183*x^4 + 2362*x^5 + 42449*x^6 + ...
where
A(x) = 1 + x*(1+x)/(1+x+x^2) + 2!*x^2*(1+x)*(2+x)/((1+x+x^2)*(1+4*x+2*x^2)) + 3!*x^3*(1+x)*(2+x)*(3+x)/((1+x+x^2)*(1+4*x+2*x^2)*(1+9*x+3*x^2)) + 4!*x^4*(1+x)*(2+x)*(3+x)*(4+x)/((1+x+x^2)*(1+4*x+2*x^2)*(1+9*x+3*x^2)*(1+16*x+4*x^2)) + ...

Links

Crossrefs

Cf. A221371, A208237, A210438.

Programs

  • Mathematica
    a[n_] := Coefficient[Sum[m!*x^m*Product[(k + x)/(1 + k^2*x + k*x^2), {k, 1, m}], {m, 0, n}] + O[x]^(n + 1), x, n]; Table[a[n], {n, 0, 18}] (* Robert P. P. McKone, Sep 15 2023 *)
  • PARI
    {a(n)=polcoeff( sum(m=0, n, m!*x^m*prod(k=1, m, (k+x)/(1+k^2*x+k*x^2 +x*O(x^n))) ), n)}
for(n=0, 30, print1(a(n), ", "))

Formula

O.g.f.: 1/(1-x*(1+x)/(1-1*2*x/(1-2*x*(2+x)/(1-2*3*x/(1-3*x*(3+x)/(1-3*4*x/(1-4*x*(4+x)/(1-4*5*x/(1-5*x*(5+x)/(1-5*6*x/(1+...))))))))))) (continued fraction).
a(n) ~ 2^(2*n+5) * n^(2*n+5/2) / (exp(2*n) * Pi^(2*n+3/2)). - Vaclav Kotesovec, Nov 02 2014

A209778 G.f.: Sum_{n>=0} x^n * Product_{k=1..n} (1 + k^2*x) / (1 + x + k^2*x^2).

Original entry on oeis.org

1, 1, 1, 3, 5, 19, 49, 203, 733, 3315, 15241, 76731, 419973, 2375027, 14842721, 94159595, 655550445, 4632480883, 35405788601, 276183156827, 2295741573013, 19588533436019, 175928886218769, 1628494746863243, 15721340742796029, 156753433757122035, 1619488446357906409
Offset: 0

Views

Author

Paul D. Hanna, Jan 19 2013

Keywords

Comments

Compare to the identity:
Sum_{n>=0} x^n * Product_{k=1..n} (1 + t*k*x) / (1 + x + t*k*x^2) = (1+x)/(1-t*x^2).

Examples

			G.f.: A(x) = 1 + x + x^2 + 3*x^3 + 5*x^4 + 19*x^5 + 49*x^6 + 203*x^7 +...
where
A(x) = 1 + x*(1+x)/(1+x+x^2) + x^2*(1+x)*(1+4*x)/((1+x+x^2)*(1+x+4*x^2)) + x^3*(1+x)*(1+4*x)*(1+9*x)/((1+x+x^2)*(1+x+4*x^2)*(1+x+9*x^2)) + x^4*(1+x)*(1+4*x)*(1+9*x)*(1+16*x)/((1+x+x^2)*(1+x+4*x^2)*(1+x+9*x^2)*(1+x+16*x^2)) +...

Links

Crossrefs

Cf. A208237, A221371.

Programs

  • PARI
    {a(n)=polcoeff( sum(m=0, n, x^m*prod(k=1, m, (1+k^2*x)/(1+x+k^2*x^2 +x*O(x^n))) ), n)}
for(n=0, 30, print1(a(n), ", "))

A221971 G.f.: A(x,y) = Sum_{n>=0} n! * x^n*y^n * Product_{k=1..n} (1 + k*x) / (1 + k*x*y + k^2*x^2*y).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 0, 4, 1, 0, 0, 3, 11, 1, 0, 0, 0, 27, 26, 1, 0, 0, 0, 17, 148, 57, 1, 0, 0, 0, 0, 278, 646, 120, 1, 0, 0, 0, 0, 155, 2590, 2481, 247, 1, 0, 0, 0, 0, 0, 4073, 18304, 8805, 502, 1, 0, 0, 0, 0, 0, 2073, 58427, 109699, 29682, 1013, 1, 0, 0, 0
Offset: 0

Views

Author

Paul D. Hanna, Feb 01 2013

Keywords

Examples

			Triangle begins:
1;
0, 1;
0, 1, 1;
0, 0, 4, 1;
0, 0, 3, 11, 1;
0, 0, 0, 27, 26, 1;
0, 0, 0, 17, 148, 57, 1;
0, 0, 0, 0, 278, 646, 120, 1;
0, 0, 0, 0, 155, 2590, 2481, 247, 1;
0, 0, 0, 0, 0, 4073, 18304, 8805, 502, 1;
0, 0, 0, 0, 0, 2073, 58427, 109699, 29682, 1013, 1;
0, 0, 0, 0, 0, 0, 80712, 614819, 590254, 96648, 2036, 1;
0, 0, 0, 0, 0, 0, 38227, 1665829, 5340996, 2948040, 307255, 4083, 1; ...

Links

Crossrefs

Cf. A208237 (row sums), A110501 (central terms), A005439 (column sums), A136126.

Programs

  • PARI
    {T(n,k)=polcoeff(polcoeff(sum(m=0, n,m!*x^m*y^m*prod(k=1, m, (1+k*x)/(1+k*x*y+k^2*x^2*y +x*O(x^n)))),n,x),k,y)}
for(n=0, 12, for(k=0, n, print1(T(n, k), ", ")); print(""))

Formula

Row sums equal A208237.
Central terms equal A110501, the Genocchi numbers of first kind (unsigned).
Columns sums equal A005439, the Genocchi numbers of second kind.

A221973 G.f.: Sum_{n>=0} n! * x^n * Product_{k=1..n} (3 + k*x)/(1 + 3*k*x + k^2*x^2).

Original entry on oeis.org

1, 3, 10, 39, 183, 1026, 6695, 49623, 411050, 3763599, 37757055, 411894882, 4854301087, 61459583007, 831926801290, 11989221944871, 183273754945959, 2961997167865410, 50462267599637975, 903853088211536295, 16980055625062979306, 333846342195447641343
Offset: 0

Views

Author

Paul D. Hanna, Feb 01 2013

Keywords

Examples

			G.f.: A(x) = 1 + 3*x + 10*x^2 + 39*x^3 + 183*x^4 + 1026*x^5 + 6695*x^6 +...
where
A(x) = 1 + x*(3+x)/(1+3*x+x^2) + 2!*x^2*(3+x)*(3+2*x)/((1+3*x+x^2)*(1+6*x+4*x^2)) + 3!*x^3*(3+x)*(3+2*x)*(3+3*x)/((1+3*x+x^2)*(1+6*x+4*x^2)*(1+9*x+9*x^2)) + 4!*x^4*(3+x)*(3+2*x)*(3+3*x)*(3+4*x)/((1+3*x+x^2)*(1+6*x+4*x^2)*(1+9*x+9*x^2)*(1+12*x+16*x^2)) +...

Crossrefs

Cf. A208237, A136127.

Programs

  • PARI
    {a(n)=polcoeff( sum(m=0, n, m!*x^m*prod(k=1, m, (3+k*x)/(1+3*k*x+k^2*x^2 +x*O(x^n))) ), n)}
for(n=0, 30, print1(a(n), ", "))

A221988 G.f.: Sum_{n>=0} n! * (2*x)^n * Product_{k=1..n} (1 + k*x)/(1 + 2*k*x + 2*k^2*x^2).

Original entry on oeis.org

1, 2, 6, 24, 116, 664, 4392, 32928, 276016, 2557856, 25965408, 286538112, 3415359296, 43727878528, 598510015104, 8720853182976, 134778021389056, 2202055694727680, 37923940767905280, 686639853639505920, 13038833241899856896, 259119925532534413312
Offset: 0

Views

Author

Paul D. Hanna, Feb 02 2013

Keywords

Examples

			G.f.: A(x) = 1 + 2*x + 6*x^2 + 24*x^3 + 116*x^4 + 664*x^5 + 4392*x^6 +...
where
A(x) = 1 + (2*x)*(1+x)/(1+2*x+2*x^2) + 2!*(2*x)^2*(1+x)*(1+2*x)/((1+2*x+2*x^2)*(1+4*x+8*x^2)) + 3!*(2*x)^3*(1+x)*(1+2*x)*(1+3*x)/((1+2*x+2*x^2)*(1+4*x+8*x^2)*(1+6*x+18*x^2)) + 4!*(2*x)^4*(1+x)*(1+2*x)*(1+3*x)*(1+4*x)/((1+2*x+2*x^2)*(1+4*x+8*x^2)*(1+6*x+18*x^2)*(1+8*x+32*x^2)) +...

Crossrefs

Cf. A221987, A208237, A136127, A221973.

Programs

  • PARI
    {a(n)=polcoeff( sum(m=0, n, m!*(2*x)^m*prod(k=1, m, (1+k*x)/(1+2*k*x+2*k^2*x^2 +x*O(x^n))) ), n)}
for(n=0, 25, print1(a(n), ", "))
Showing 1-5 of 5 results.

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