A325046 - OEIS

%I #15 Jun 11 2022 11:42:24

%S 1,2,3,4,6,8,9,16,16,18,36,34,27,68,76,58,86,122,170,176,99,206,436,

%T 350,192,392,574,690,840,730,657,804,1328,2218,2070,846,910,2794,4012,

%U 3818,3306,3176,4109,4280,4546,8550,11694,9366,5726,5016,8338,15636,23498,24736,16434,8474,14423,28616,32114,31256,42116,51828,50476,42378,28306,26454,56358,101900,133758,132356,87490,41024,53475,109392,158936,190868,232342,265698,221026,158178,200048,269954,239516,206696,314724,516784,710010,774678,576170,255094,134523

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

%C Odd terms occur only at positions n*(n+1) for n >= 0 (conjecture).

%H Paul D. Hanna, <a href="/A325046/b325046.txt">Table of n, a(n) for n = 0..10100</a>

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

%F G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k) * (x^n + x^k)^(n-k).

%F G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k) * Sum_{j=0..n-k} binomial(n-k,j) * x^((n-k)*(n-j)).

%e G.f.: A(x) = 1 + 2*x + 3*x^2 + 4*x^3 + 6*x^4 + 8*x^5 + 9*x^6 + 16*x^7 + 16*x^8 + 18*x^9 + 36*x^10 + 34*x^11 + 27*x^12 + 68*x^13 + 76*x^14 + 58*x^15 + 86*x^16 + 122*x^17 + 170*x^18 + 176*x^19 + 99*x^20 + 206*x^21 + 436*x^22 + 350*x^23 + 192*x^24 + 392*x^25 + 574*x^26 + 690*x^27 + 840*x^28 + 730*x^29 + 657*x^30 + 804*x^31 + 1328*x^32 + 2218*x^33 + 2070*x^34 + 846*x^35 + 910*x^36 + 2794*x^37 + 4012*x^38 + 3818*x^39 + 3306*x^40 + 3176*x^41 + 4109*x^42 + ...

%e such that

%e A(x) = 1/(1 - x) + x*(1 + x)/(1 - x^2)^2 + x^2*(1 + x^2)^2/(1 - x^3)^3 + x^3*(1 + x^3)^3/(1 - x^4)^4 + x^4*(1 + x^4)^4/(1 - x^5)^5 + x^5*(1 + x^5)^5/(1 - x^6)^6 + x^6*(1 + x^6)^6/(1 - x^7)^7 + x^7*(1 + x^7)^7/(1 - x^8)^8 + ...

%e ODD TERMS.

%e It appears that odd terms occur only at n*(n+1); the odd terms begin:

%e [1, 3, 9, 27, 99, 657, 4109, 14423, 53475, 134523, 1686983, 13421711, 85848955, 325004679, 1482972731, 6258674687, 43509358107, ..., A325047(n), ...].

%e The terms at positions n*(n+2), for n >= 0, start as:

%e [1, 4, 16, 58, 192, 846, 5726, 42378, 200048, 816738, 1924336, 10968450, 79124014, 854427564, 4293474170, 23269170810, 100555730012, 543827171600, ...].

%e TRIANGLE FORM.

%e This sequence may be written as a triangle like so

%e 1, 2;

%e 3, 4, 6, 8;

%e 9, 16, 16, 18, 36, 34;

%e 27, 68, 76, 58, 86, 122, 170, 176;

%e 99, 206, 436, 350, 192, 392, 574, 690, 840, 730;

%e 657, 804, 1328, 2218, 2070, 846, 910, 2794, 4012, 3818, 3306, 3176;

%e 4109, 4280, 4546, 8550, 11694, 9366, 5726, 5016, 8338, 15636, 23498, 24736, 16434, 8474;

%e 14423, 28616, 32114, 31256, 42116, 51828, 50476, 42378, 28306, 26454, 56358, 101900, 133758, 132356, 87490, 41024;

%e 53475, 109392, 158936, 190868, 232342, 265698, 221026, 158178, 200048, 269954, 239516, 206696, 314724, 516784, 710010, 774678, 576170, 255094; ...

%e in which the odd terms form the leftmost border.

%o (PARI) {a(n) = my(A=sum(m=0, n, x^m * (1 + x^m +x*O(x^n))^m/(1 - x^(m+1) +x*O(x^n))^(m+1) )); polcoeff(A, n)}

%o for(n=0, 120, print1(a(n), ", "))

%Y Cf. A325047 (odd terms), A323557 (variant).

%K nonn

%O 0,2

%A _Paul D. Hanna_, Mar 26 2019