A392528 G.f. Sum_{n>=0} x^n / (1 - x)^A002262(n), where A002262 is a triangle read by rows in which row n lists the first n+1 nonnegative integers.

1, 1, 1, 2, 2, 3, 5, 6, 8, 11, 16, 21, 28, 37, 49, 66, 87, 115, 151, 197, 256, 333, 431, 558, 721, 929, 1193, 1527, 1950, 2483, 3157, 4008, 5081, 6431, 8125, 10245, 12893, 16192, 20299, 25406, 31751, 39627, 49393, 61487, 76442, 94907, 117668, 145686, 180129, 222421, 274297, 337869, 415704, 510916, 627275

Offset

0,4

Links

Paul D. Hanna, Table of n, a(n) for n = 0..5050

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n may be expressed as follows.

(1) A(x) = Sum_{n>=0} x^n / (1 - x)^A002262(n).

(2) A(x) = Sum_{n>=0} x^n/(1-x)^n * Sum_{k >= n} x^(k*(k+1)/2).

(3) A(x) = (1-x)/(1-2*x) * T(x) - Sum_{n>=1} x^n/(1-x)^n * Sum_{k=0..n-1} x^(k*(k+1)/2), where T(q) = theta_2(q^(1/2)) / (2*q^(1/8)).

Example

G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 2*x^4 + 3*x^5 + 5*x^6 + 6*x^7 + 8*x^8 + 11*x^9 + 16*x^10 + 21*x^11 + 28*x^12 + 37*x^13 + 49*x^14 + 66*x^15 + 87*x^16 + 115*x^17 + 151*x^18 + 197*x^19 + 256*x^20 + ...
where A(x) = Sum_{n>=0} x^n/(1 - x)^A002262(n) is illustrated by
A(x) = 1 +
  x + x^2/(1-x) +
  x^3 + x^4/(1-x) + x^5/(1-x)^2 +
  x^6 + x^7/(1-x) + x^8/(1-x)^2 + x^9/(1-x)^3 +
  x^10 + x^11/(1-x) + x^12/(1-x)^2 + x^13/(1-x)^3 + x^14/(1-x)^4 +
  x^15 + x^16/(1-x) + x^17/(1-x)^2 + x^18/(1-x)^3 + x^19/(1-x)^4 + x^20/(1-x)^5 + ...
which may be rewritten as
A(x) = (1 + x + x^3 + x^6 + x^10 + x^15 + ...) +
  x/(1-x) * (x + x^3 + x^6 + x^10 + x^15 + ...) +
  x^2/(1-x)^2 * (x^3 + x^6 + x^10 + x^15 + ...) +
  x^3/(1-x)^3 * (x^6 + x^10 + x^15 + x^21 + ...) +
  x^4/(1-x)^4 * (x^10 + x^15 + x^21 + x^28 + ...) + ...,
a sum involving truncated series expansions for theta_2(x^(1/2))/(2*x^(1/8)) = 1 + x + x^3 + x^6 + x^10 + x^15 + ... + x^(k*(k+1)/2) + ...
SPECIFIC VALUES.
A(1/2) = Sum_{n>=0} (n+1) / 2^(n*(n+1)/2) = 2.4425692857624315691939233231717658729567434832591...
A(1/3) = Sum_{n>=0} (2^(n+1) - 1)/(2^n * 3^(n*(n+1)/2)) = 1.5674197804037130142415947955218985549709251892979...
A(1/4) = Sum_{n>=0} (3^(n+1) - 1)/(2*3^n * 4^(n*(n+1)/2)) = 1.35626589361259719134004326703004157334951706065645...
A(1/5) = Sum_{n>=0} (4^(n+1) - 1)/(3*4^n * 5^(n*(n+1)/2)) = 1.2605851364436827960677909396256663684561830385555...
A(t) = 2 at t = 0.4378287625483057432427479772426607930430743119...
A(t) = 3 at t = 0.5489627750246421219203474323241407265689887623...
A(t) = 4 at t = 0.6001769018172804281709767500838134223639193622...
A(t) = 5 at t = 0.6303589650297074855750910364988318656676429507...

Prog

(PARI) \\ G.f.: Sum_{n>=0} x^n/(1 - x)^A002262(n)
{A002024(n) = floor(sqrt(2*n) + 1/2)}
{A002262(n) = n - A002024(n+1) * (A002024(n+1) - 1)/2}
{a(n) = my(A, x = 'x +x*O(x^n));
A = sum(m=0, n, x^m / (1 - x)^A002262(m) ); polcoef(A, n)}
for(n=0, 60, print1(a(n), ", "))
(PARI) \\ G.f.: Sum_{n>=0} x^n/(1-x)^n * Sum_{k >= n} x^(k*(k+1)/2)
{A002024(n) = floor(sqrt(2*n) + 1/2)}
{a(n) = my(A, x = 'x +x*O(x^n));
A = sum(m=0, n, x^m/(1 - x)^m * sum(k=m, A002024(n+1), x^(k*(k+1)/2) ) ); polcoef(A, n)}
for(n=0, 60, print1(a(n), ", "))

Crossrefs

Cf. A392529, A002024, A002262, A089799.

Sequence in context: A118399 A278298 A178927 * A076571 A084783 A265853

Adjacent sequences: A392525 A392526 A392527 * A392529 A392530 A392531

Keyword

nonn

Author

Paul D. Hanna, Feb 01 2026

Status

approved