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A340942 Row k = 1 of rectangular table A340940. 10

%I #15 Mar 11 2021 01:58:56

%S 1,1,2,9,46,253,1467,8842,54878,348489,2254007,14799922,98405915,

%T 661315560,4485060016,30660329139,211061833430,1461899228702,

%U 10181435691655,71258997482752,500958491353507,3536041471033681,25051475034240297

%N Row k = 1 of rectangular table A340940.

%C The g.f. A(x) of this sequence employs the related identities:

%C (1) Sum_{n>=0} (n+1) * p^n/(1 - q*r^n) = Sum_{n>=0} q^n/(1 - p*r^n)^2,

%C (2) Sum_{n>=0} (n+1) * p^n/(1 - q*r^n)^2 = Sum_{n>=0} (n+1) * q^n/(1 - p*r^n)^2.

%H Vaclav Kotesovec, <a href="/A340942/b340942.txt">Table of n, a(n) for n = 0..200</a>

%F G.f. A(x) satisfies:

%F (1) A(x) = P(x)/Q(x) where

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

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

%F (2) A(x) = P(x)/Q(x) where

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

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

%e G.f. A(x) = 1 + x + 2*x^2 + 9*x^3 + 46*x^4 + 253*x^5 + 1467*x^6 + 8842*x^7 + 54878*x^8 + 348489*x^9 + 2254007*x^10 + 14799922*x^11 + 98405915*x^12 + ...

%e such that A(x) = P(x)/Q(x) where

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

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

%e equivalently,

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

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

%e explicitly,

%e P(x) = 1 + 4*x + 14*x^2 + 54*x^3 + 241*x^4 + 1214*x^5 + 6651*x^6 + 38566*x^7 + 232727*x^8 + 1446432*x^9 + 9196742*x^10 + 59545914*x^11 + 391304285*x^12 + ...

%e Q(x) = 1 + 3*x + 9*x^2 + 30*x^3 + 120*x^4 + 562*x^5 + 2939*x^6 + 16523*x^7 + 97551*x^8 + 596461*x^9 + 3744416*x^10 + 23996814*x^11 + 156370334*x^12 + ...

%o (PARI) {a(n) = my(A=1+x+x*O(x^n),P=1,Q=1);

%o for(i=0,n,

%o P = sum(m=0,n, (m+1)*x^m*A^(2*m)/(1 - x*A^m + x*O(x^n))^2 );

%o Q = sum(m=0,n, (m+1)*x^m*A^m/(1 - x*A^(m+1) + x*O(x^n)) );

%o A = P/Q); polcoeff(A,n)}

%o for(n=0,20, print1(a(n),", "))

%o (PARI) {a(n) = my(A=1+x+x*O(x^n),P=1,Q=1);

%o for(i=0,n,

%o P = sum(m=0,n, (m+1)*x^m/(1 - x*A^(m+2) + x*O(x^n))^2 );

%o Q = sum(m=0,n, x^m*A^m/(1 - x*A^(m+1) + x*O(x^n))^2 );

%o A = P/Q); polcoeff(A,n)}

%o for(n=0,20, print1(a(n),", "))

%o (PARI) /* As generated by Finite Differences in the columns of table A340940 */

%o {A340940(k,n) = my(A=[1,1]); for(i=1,n, A=concat(A,0); H=A; A=concat(A,0);

%o H[#A-1] = -polcoeff( sum(m=0,#A, x^m/(1 - x*Ser(A)^(m+k)) ) - sum(m=0,#A, x^m*Ser(A)^m/(1 - x*Ser(A)^(k*m+k-1)) ), #A)/(k-1); A=H); A[n+1] }

%o for(n=0,20, if(n<2,print1("1, "), print1( Vec(-(x-1)^n*Ser(vector(n+1,k,A340940(k+1,n))))[n],", ") ))

%Y Cf. A340940, A340941, A340894, A340895.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Feb 03 2021

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Last modified March 29 04:23 EDT 2024. Contains 371264 sequences. (Running on oeis4.)