A392528 - OEIS

%I #27 Feb 05 2026 16:35:41

%S 1,1,1,2,2,3,5,6,8,11,16,21,28,37,49,66,87,115,151,197,256,333,431,

%T 558,721,929,1193,1527,1950,2483,3157,4008,5081,6431,8125,10245,12893,

%U 16192,20299,25406,31751,39627,49393,61487,76442,94907,117668,145686,180129,222421,274297,337869,415704,510916,627275

%N 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.

%H Paul D. Hanna, <a href="/A392528/b392528.txt">Table of n, a(n) for n = 0..5050</a>

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

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

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

%F (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)).

%e 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 + ...

%e where A(x) = Sum_{n>=0} x^n/(1 - x)^A002262(n) is illustrated by

%e A(x) = 1 +

%e   x + x^2/(1-x) +

%e   x^3 + x^4/(1-x) + x^5/(1-x)^2 +

%e   x^6 + x^7/(1-x) + x^8/(1-x)^2 + x^9/(1-x)^3 +

%e   x^10 + x^11/(1-x) + x^12/(1-x)^2 + x^13/(1-x)^3 + x^14/(1-x)^4 +

%e   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 + ...

%e which may be rewritten as

%e A(x) = (1 + x + x^3 + x^6 + x^10 + x^15 + ...) +

%e   x/(1-x) * (x + x^3 + x^6 + x^10 + x^15 + ...) +

%e   x^2/(1-x)^2 * (x^3 + x^6 + x^10 + x^15 + ...) +

%e   x^3/(1-x)^3 * (x^6 + x^10 + x^15 + x^21 + ...) +

%e   x^4/(1-x)^4 * (x^10 + x^15 + x^21 + x^28 + ...) + ...,

%e 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) + ...

%e SPECIFIC VALUES.

%e A(1/2) = Sum_{n>=0} (n+1) / 2^(n*(n+1)/2) = 2.4425692857624315691939233231717658729567434832591...

%e A(1/3) = Sum_{n>=0} (2^(n+1) - 1)/(2^n * 3^(n*(n+1)/2)) = 1.5674197804037130142415947955218985549709251892979...

%e A(1/4) = Sum_{n>=0} (3^(n+1) - 1)/(2*3^n * 4^(n*(n+1)/2)) = 1.35626589361259719134004326703004157334951706065645...

%e A(1/5) = Sum_{n>=0} (4^(n+1) - 1)/(3*4^n * 5^(n*(n+1)/2)) = 1.2605851364436827960677909396256663684561830385555...

%e A(t) = 2 at t = 0.4378287625483057432427479772426607930430743119...

%e A(t) = 3 at t = 0.5489627750246421219203474323241407265689887623...

%e A(t) = 4 at t = 0.6001769018172804281709767500838134223639193622...

%e A(t) = 5 at t = 0.6303589650297074855750910364988318656676429507...

%o (PARI) \\ G.f.: Sum_{n>=0} x^n/(1 - x)^A002262(n)

%o {A002024(n) = floor(sqrt(2*n) + 1/2)}

%o {A002262(n) = n - A002024(n+1) * (A002024(n+1) - 1)/2}

%o {a(n) = my(A,x = 'x +x*O(x^n));

%o A = sum(m=0,n, x^m / (1 - x)^A002262(m) ); polcoef(A,n)}

%o for(n=0,60,print1(a(n),", "))

%o (PARI) \\ G.f.: Sum_{n>=0} x^n/(1-x)^n * Sum_{k >= n} x^(k*(k+1)/2)

%o {A002024(n) = floor(sqrt(2*n) + 1/2)}

%o {a(n) = my(A,x = 'x +x*O(x^n));

%o A = sum(m=0,n, x^m/(1 - x)^m * sum(k=m,A002024(n+1), x^(k*(k+1)/2) ) ); polcoef(A,n)}

%o for(n=0,60,print1(a(n),", "))

%Y Cf. A392529, A002024, A002262, A089799.

%K nonn

%O 0,4

%A _Paul D. Hanna_, Feb 01 2026