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A260687 Triangular array with n-th row giving coefficients of polynomial Product_{k = 2..n} (k + n*t) for n >= 1. 3

%I #9 Nov 17 2015 21:02:18

%S 1,2,2,6,15,9,24,104,144,64,120,770,1775,1750,625,720,6264,20880,

%T 33480,25920,7776,5040,56196,250096,571095,708295,453789,117649,40320,

%U 554112,3127040,9433088,16486400,16744448,9175040,2097152,362880,5973264,41229324,156498804

%N Triangular array with n-th row giving coefficients of polynomial Product_{k = 2..n} (k + n*t) for n >= 1.

%C Related to A220883 and A251592.

%D R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, Reading, MA, 2nd ed. 1998

%H P. Bala, <a href="/A251592/a251592.pdf">Fractional iteration of a series inversion operator</a>

%F E.g.f. (with constant term 1 included): A(x,t) = [ 1/x*Revert( x*(1 - x)^t ) ]^(1/t) = Sum_{n >= 0} 1/(n*t + 1)*binomial(n*t + n,n)*x^n = 1 + x + (2 + 2*t)*x^2/2! + (2 + 3*t)*(3 + 3*t)*x^3/3! + (2 + 4*t)*(3 + 4*t)*(4 + 4*t)*x^4/4! + ..., where Revert denotes the series reversion operator with respect to x.

%F In the notation of the Bala link, A(x,t) = I^t(1/(1 - x)) where I^t is a fractional inversion operator.

%F A(x,t) = B_(1+t)(x), where B_t(x) is the e.g.f. for A251592 and is the generalized binomial series of Lambert. See Graham et al., Section 5.4 and Section 7.5.

%F A(x,t)^m = Sum_{n >= 0} m/(n*t + m)*binomial(n*t + n + m - 1,n)*x^n = 1 + m*x + m*(2*t + m + 1)*x^2/2! + m*(3*t + m + 1)*(3*t + m + 2)*x^3/3! + m*(4*t + m + 1)*(4*t + m + 2)*(4*t + m + 3)*x^4/4! + ....

%F A(x,t)^t = 1 + t*x + t(1 + 3*t)*x^2/2! + t*(1 + 4*t)*(2 + 4*t)*x^3/3! + t*(1 + 5*t)*(2 + 5*t)*(3 + 5*t)*x^4/4! + ... is the e.g.f for A220883 with an extra constant term 1 and an extra factor of t included.

%F t*log( A(x,t) ) = t*x + t*(1 + 2*t)*x^2/2! + t*(1 + 3*t)*(2 + 3*t)*x^3/3! + t*(1 + 4*t)*(2 + 4*t)*(3 + 4*t)*x^4/4! + ... is the e.g.f for A056856.

%F For n = 1,2,3,..., the sequence [x^n] A(x,t)^n = [1, (2*t + 3), (3*t + 4)*(3*t + 5)/2!, (4*t + 5)*(4*t + 6)*(4*t + 7)/3!, ...]. This sequence has the following specializations:

%F t = 0: [1, 3, 10, 35, 126, ...] = A001700 (with different offset).

%F t = 1: [1, 5, 28, 165, 1001, ...] = A025174.

%F t = 2: [1, 7, 55, 455, 3876, ...] = A224274.

%F t = 3: [1, 9, 91, 969, 10626, ...] = A163456.

%e Triangle begins

%e ...1

%e ...2 2

%e ...6 15 9

%e ..24 104 144 64

%e .120 770 1775 1750 625

%e .720 6264 20880 33480 25920 7776

%e 5040 56196 250096 571095 708295 453789 117649

%e ...

%p seq(seq(coeff(mul(n*t + k, k = 2 .. n), t, i), i = 0..n-1), n = 1..10);

%Y A000142 (column 0), A000169 (main diagonal), A006675 (column 1). Cf. A001700, A025174, A056856, A163456, A220883, A224274, A251592.

%K nonn,tabl,easy

%O 1,2

%A _Peter Bala_, Nov 16 2015

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Last modified March 28 18:04 EDT 2024. Contains 371254 sequences. (Running on oeis4.)