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 1-4 of 4 results.

A039924 G.f.: Sum_{k>=0} x^(k^2)*(-1)^k/(Product_{i=1..k} 1-x^i).

Original entry on oeis.org

1, -1, -1, -1, 0, 0, 1, 1, 2, 1, 2, 1, 1, 0, 0, -2, -1, -3, -3, -4, -3, -5, -3, -4, -2, -3, 0, -1, 3, 2, 5, 5, 9, 7, 11, 9, 13, 10, 13, 9, 12, 7, 9, 3, 5, -3, -1, -9, -7, -17, -15, -24, -21, -31, -27, -37, -31, -40, -33, -41, -31, -39, -27, -33, -18, -24, -6, -11, 9, 5, 26, 23
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Ramanujan used the form Sum_{k>=0} x^(k^2) / (Product_{i=1..k} 1-(-x)^i), which is obtained by changing the sign of x. - Michael Somos, Jul 20 2003
Coefficients in expansion of determinant of infinite tridiagonal matrix shown below in powers of x^2 (Lehmer 1973):
1 x 0 0 0 0 ...
x 1 x^2 0 0 0 ...
0 x^2 1 x^3 0 0 ...
0 0 x^3 1 x^4 0 ...
... ... ... ... ... ... ...
Convolution inverse of A003116.

Examples

			G.f. = 1 - x - x^2 - x^3 + x^6 + x^7 + 2*x^8 + x^9 + 2*x^10 + x^11 + x^12 + ...

References

  • N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 55, Eq. (26.11).
  • G. H. Hardy, P. V. Seshu Aiyar and B. M. Wilson, editors, Collected Papers of Srinivasa Ramanujan, Cambridge, 1923; p. 354.
  • D. H. Lehmer, Combinatorial and cyclotomic properties of certain tridiagonal matrices. Proceedings of the Fifth Southeastern Conference on Combinatorics, Graph Theory, and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1974), pp. 53-74. Congressus Numerantium, No. X, Utilitas Math., Winnipeg, Man., 1974. MR0441852.
  • Herman P. Robinson, personal communication to N. J. A. Sloane.

Links

Crossrefs

Cf. A003116, A224898.

Programs

  • Maple
    qq:=n->mul( 1-(-q)^i, i=1..n); add (q^(n^2)/qq(n),n=0..100): series(t1,q,99);
  • Mathematica
    CoefficientList[ Series[ Sum[x^k^2*(-1)^k / Product[1-x^i,{i,1,k}], {k,0,100}], {x,0,100}],x][[1 ;; 72]] (* Jean-François Alcover, Apr 08 2011 *)
a[ n_] := If[n < 0, 0, SeriesCoefficient[ Sum[ (-1)^k x^k^2 / QPochhammer[ x, x, k], {k, 0, Sqrt[n]}], {x, 0, n}]] (* Michael Somos, Jan 04 2014 *)
  • PARI
    {a(n) = if( n<0, 0, polcoeff( sum(k=0, sqrtint(n), x^k^2 / prod(i=1, k, x^i - 1, 1 + x * O(x^n))), n))} /* Michael Somos, Jul 20 2003 */

Formula

a(n) = A286041(n) - A286316(n) (conjectured). - George Beck, May 05 2017
Proof from Doron Zeilberger, Aug 20 2018: (Start)
The generating function for partitions whose parts differ by at least 2 with exactly k parts is (famously) q^(k^2)/((1-q)*...*(1-q^k)).
Indeed, if you take any such partition and remove 1 from the smallest part, 3 from the second-smallest part, etc., you remove 1+3+...+(2k-1) = k^2 and are left with an ordinary partition whose number of parts is <= k whose generating function is 1/((1-q)*...*(1-q)^k).
Summing these up famously gives the generating function for partitions whose differences is >= 2. Sticking a (-1)^k in front gives the generating function for the difference between such partitions with an even number of parts and an odd number of parts, since (-1)^even=1 and (-1)^odd=-1. (End)

Extensions

More terms from Vladeta Jovovic, Mar 05 2001

A286744 Number of distinct partitions of n with parts differing by at least two with smallest part at least two and with an even number of parts.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 9, 9, 11, 12, 15, 16, 20, 22, 27, 30, 36, 40, 48, 53, 62, 69, 80, 88, 101, 111, 126, 138, 156, 170, 191, 208, 232, 253, 282, 306, 340, 370, 410, 446, 494, 537, 594, 647, 714, 778, 859, 935, 1031, 1124
Offset: 0

Views

Author

George Beck, May 13 2017

Keywords

Examples

			a(8) = 2 because 6+2 and 5+3 are the only partitions of 8 that satisfy the three conditions.

Links

Crossrefs

Cf. A286745, A003106, A224898, A001622.

Programs

  • Maple
    b:= proc(n, i, t) option remember; `if`(n=0, t,
      `if`(i>n, 0, b(n, i+1, t)+b(n-i, i+2, 1-t)))
    end:
a:= n-> b(n, 2, 1):
seq(a(n), n=0..80);  # Alois P. Heinz, Nov 23 2017
  • Mathematica
    Table[Length@
  Select[IntegerPartitions@n,
   Min[-Differences@#] >= 2 && Min@# >= 2 && EvenQ@Length@# &], {n,
  20}]
(* Second program: *)
b[n_, i_, t_] := b[n, i, t] = If[n == 0, t,
     If[i > n, 0, b[n, i + 1, t] + b[n - i, i + 2, 1 - t]]];
a[n_] := b[n, 2, 1];
a /@ Range[0, 80] (* Jean-François Alcover, Jun 06 2021, after Alois P. Heinz *)

Formula

a(n) ~ exp(2*Pi*sqrt(n/15)) / (4 * 3^(1/4) * sqrt(5*phi) * n^(3/4)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Mar 10 2020

A286745 Number of distinct partitions of n with parts differing by at least two with smallest part at least two and with an odd number of parts.

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 9, 11, 13, 15, 17, 20, 22, 25, 28, 31, 34, 39, 42, 47, 52, 58, 64, 72, 79, 89, 99, 111, 123, 139, 154, 173, 193, 216, 240, 269, 298, 333, 369, 410, 453, 503, 554, 613, 674, 743, 815, 897, 981, 1077, 1177, 1288, 1405, 1536, 1672, 1825, 1985, 2163, 2350, 2558, 2776, 3019, 3275, 3557, 3856, 4186, 4534, 4919
Offset: 0

Views

Author

George Beck, May 13 2017

Keywords

Examples

			a(12) = 2 because of the partitions of 12, 12 and 6+4+2 are the only two that satisfy all three conditions.

Links

Crossrefs

Cf. A286744, A003106, A224898.

Programs

  • Maple
    b:= proc(n, i, t) option remember; `if`(n=0, t,
      `if`(i>n, 0, b(n, i+1, t)+b(n-i, i+2, 1-t)))
    end:
a:= n-> b(n, 2, 0):
seq(a(n), n=0..80);  # Alois P. Heinz, Nov 23 2017
  • Mathematica
    Table[Length@ Select[ip@n, Min[-Differences@#] >= 2 && Min@# >= 2 && OddQ@Length@# &], {n, 20}]

Formula

a(n) ~ exp(2*Pi*sqrt(n/15)) / (4 * 3^(1/4) * sqrt(5*phi) * n^(3/4)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Mar 10 2020

A227620 Logarithmic derivative of A005169, the number of fountains of n coins.

Original entry on oeis.org

1, 1, 4, 5, 11, 22, 36, 69, 121, 221, 386, 686, 1210, 2122, 3734, 6517, 11408, 19903, 34714, 60485, 105312, 183272, 318758, 554262, 963361, 1674076, 2908426, 5052066, 8774386, 15237482, 26458718, 45939797, 79759442, 138468656, 240382216, 417289619, 724369536, 1257396992
Offset: 1

Views

Author

Paul D. Hanna, Jul 17 2013

Keywords

Examples

			L.g.f.: L(x) = x + x^2/2 + 4*x^3/3 + 5*x^4/4 + 11*x^5/5 + 22*x^6/6 +...
such L(x) = log(P(x)) - log(Q(x)) where
P(x) = 1 - x^2 - x^3 - x^4 - x^5 + x^8 + x^9 + 2*x^10 + 2*x^11 + 2*x^12 + 2*x^13 + 2*x^14 + x^15 + x^16 - x^18 +...+ A224898(n)*x^n +...
Q(x) = 1 - x - x^2 - x^3 + x^6 + x^7 + 2*x^8 + x^9 + 2*x^10 + x^11 + x^12 - 2*x^15 - x^16 - 3*x^17 - 3*x^18 +...+ A039924(n)*x^n +...
log(P(x)) = -2*x^2/2 - 3*x^3/3 - 6*x^4/4 - 10*x^5/5 - 11*x^6/6 - 21*x^7/7 - 22*x^8/8 - 39*x^9/9 - 42*x^10/10 +...
log(Q(x)) = -x - 3*x^2/2 - 7*x^3/3 - 11*x^4/4 - 21*x^5/5 - 33*x^6/6 - 57*x^7/7 - 91*x^8/8 - 160*x^9/9 - 263*x^10/10 +...

Crossrefs

Cf. A227543, A005169, A039924, A224898.

Programs

  • PARI
    /* As the log of a continued fraction: */
{a(n)=local(A=x, CF=1+x); for(k=0, n, CF=1/(1-x^(n-k+1)*CF+x*O(x^n)); A=log(CF)); n*polcoeff(A, n)}
for(n=1,40,print1(a(n),", "))
  • PARI
    /* By the Rogers-Ramanujan continued fraction identity: */
{a(n)=local(A=x, P=1+x, Q=1);
P=sum(m=0, sqrtint(n), (-1)^m*x^(m*(m+1))/prod(k=1, m, 1-x^k));
Q=sum(m=0, sqrtint(n), (-1)^m*x^(m^2)/prod(k=1, m, 1-x^k));
A=log(P/(Q+x*O(x^n))); n*polcoeff(A, n)}
for(n=1,40,print1(a(n),", "))

Formula

L.g.f.: log( 1/(1-x/(1-x^2/(1-x^3/(1-x^4/(1-x^5/(1-...)))))) ), the logarithm of a continued fraction.
L.g.f.: log( P(x) / Q(x) ) where
P(x) = Sum_{n>=0} (-1)^n* x^(n*(n+1)) / Product_{k=1..n} (1-x^k),
Q(x) = Sum_{n>=0} (-1)^n* x^(n^2) / Product_{k=1..n} (1-x^k),
due to the Rogers-Ramanujan continued fraction identity.
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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