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A384832 G.f. A(x) = Sum_{n>=0} x^n * Product_{k=0..n} ((1+x)^(n-k+1) - x^k).

%I A384832 #9 Jun 30 2025 10:17:28
%S A384832 1,2,4,13,41,144,533,2072,8463,36142,160852,744491,3576342,17796825,
%T A384832 91587499,486686277,2666612930,15045088274,87301643726,520416443472,
%U A384832 3183640482658,19967208261651,128273336978302,843360769602607,5670286993205471,38957428760628861,273318099568893757,1956848333035887861
%N A384832 G.f. A(x) = Sum_{n>=0} x^n * Product_{k=0..n} ((1+x)^(n-k+1) - x^k).
%H A384832 Paul D. Hanna, <a href="/A384832/b384832.txt">Table of n, a(n) for n = 1..300</a>
%e A384832 G.f.: A(x) = x + 2*x^2 + 4*x^3 + 13*x^4 + 41*x^5 + 144*x^6 + 533*x^7 + 2072*x^8 + 8463*x^9 + 36142*x^10 + 160852*x^11 + 744491*x^12 + ...
%e A384832 where
%e A384832 A(x) = 1 * ((1+x) - 1) +
%e A384832   x * ((1+x)^2 - 1)*((1+x) - x) +
%e A384832   x^2 * ((1+x)^3 - 1)*((1+x)^2 - x)*((1+x) - x^2) +
%e A384832   x^3 * ((1+x)^4 - 1)*((1+x)^3 - x)*((1+x)^2 - x^2)*((1+x) - x^3) +
%e A384832   x^4 * ((1+x)^5 - 1)*((1+x)^4 - x)*((1+x)^3 - x^2)*((1+x)^2 - x^3)*((1+x) - x^4) +
%e A384832   x^5 * ((1+x)^6 - 1)*((1+x)^5 - x)*((1+x)^4 - x^2)*((1+x)^3 - x^3)*((1+x)^2 - x^4)*((1+x) - x^5) +
%e A384832   x^6 * ((1+x)^7 - 1)*((1+x)^6 - x)*((1+x)^5 - x^2)*((1+x)^4 - x^3)*((1+x)^3 - x^4)*((1+x)^2 - x^5)*((1+x) - x^6) + ...
%e A384832 equivalently,
%e A384832 A(x) = x +
%e A384832   (2*x^2 + x^3) +
%e A384832   (3*x^3 + 9*x^4 + 10*x^5 + 5*x^6 - 2*x^7 - 3*x^8 - x^9) +
%e A384832   (4*x^4 + 26*x^5 + 78*x^6 + 139*x^7 + 147*x^8 + 73*x^9 - 25*x^10 - 65*x^11 - 45*x^12 - 15*x^13 - 2*x^14) +
%e A384832   (5*x^5 + 55*x^6 + 290*x^7 + 965*x^8 + 2226*x^9 + 3689*x^10 + 4378*x^11 + 3463*x^12 + 1184*x^13 - 1161*x^14 - 2296*x^15 - 2002*x^16 - 1034*x^17 - 239*x^18 + 85*x^19 + 102*x^20 + 44*x^21 + 10*x^22 + x^23) +
%e A384832   (6*x^6 + 99*x^7 + 794*x^8 + 4099*x^9 + 15185*x^10 + 42667*x^11 + 93837*x^12 + 164301*x^13 + 229972*x^14 + 253682*x^15 + 208380*x^16 + 100483*x^17 - 28293*x^18 - 125093*x^19 - 157729*x^20 - 130285*x^21 - 73656*x^22 - 21858*x^23 + 7068*x^24 + 14241*x^25 + 10381*x^26 + 4903*x^27 + 1605*x^28 + 355*x^29 + 48*x^30 + 3*x^31) + ...
%o A384832 (PARI) {a(n) = my(A = sum(m=0,n, x^m * prod(k=0,m, (1+x)^(m-k+1) - x^k +x*O(x^n)) )); polcoef(A,n)}
%o A384832 for(n=1,30,print1(a(n),", "))
%Y A384832 Cf. A121690.
%K A384832 nonn
%O A384832 1,2
%A A384832 _Paul D. Hanna_, Jun 29 2025