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A316349 Consider coefficients U(m,l,k) defined by the identity Sum_{k=1..l} Sum_{j=0..m} A302971(m,j)/A304042(m,j) * k^j * (T-k)^j = Sum_{k=0..m} (-1)^(m-k) * U(m,l,k) * T^k that holds for all positive integers l,m,T. This sequence gives 3-column table read by rows, where the n-th row lists coefficients U(2,n,k) for k = 0, 1, 2; n >= 1. 7
31, 60, 30, 512, 540, 150, 2943, 2160, 420, 10624, 6000, 900, 29375, 13500, 1650, 68256, 26460, 2730, 140287, 47040, 4200, 263168, 77760, 6120, 459999, 121500, 8550, 760000, 181500, 11550, 1199231, 261360, 15180, 1821312, 365040, 19500, 2678143, 496860, 24570, 3830624, 661500, 30450 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

For l=T, the identity takes form T^(2m+1) = Sum_{k=0..m} (-1)^(m-k)*U(m,T,k)*T^k, which holds for all positive integers T and m.

LINKS

Table of n, a(n) for n=1..42.

Max Alekseyev, Derivation of the general formula for U(m,n,k), MathOverflow, 2018.

Petro Kolosov, Series Representation of Power Function, arXiv:1603.02468 [math.NT], 2016-2018.

Petro Kolosov, More details on derivation of present sequence.

Petro Kolosov, Mathematica program, verifies the identity T^(2m+1) = Sum_{k=0..m} (-1)^(m-k)*U(m,T,k)*T^k for m=0,1,...,12.

FORMULA

U(2,n,0) = 6*n^5 + 15*n^4 + 10*n^3; U(2,n,1) = 15*n^4 + 30*n^3 + 15*n^2; U(2,n,2) = 10*n^3 + 15*n^2 + 5*n. - Max Alekseyev, Sep 06 2018

From Colin Barker, Jul 06 2018: (Start)

G.f.: x*(31 + 60*x + 30*x^2 + 326*x^3 + 180*x^4 - 30*x^5 + 336*x^6 - 180*x^7 - 30*x^8 + 26*x^9 - 60*x^10 + 30*x^11 + x^12) / ((1 - x)^6*(1 + x + x^2)^6).

a(n) = 6*a(n-3) - 15*a(n-6) + 20*a(n-9) - 15*a(n-12) + 6*a(n-15) - a(n-18) for n>18.

(End)

U(m,l,t) = (-1)^m * Sum_{k=1..l} Sum_{j=t..m} binom(j,t) * R(m,j) * k^{2j-t} * (-1)^j, where m = 1, l >= 1 and R(m,j) = A302971(m,j)/A304042(m,j); after Max Alekseyev, see links.

EXAMPLE

       column   column  column

   l     k=0      k=1     k=2

  --  -------  -------  ------

   1       31       60      30

   2      512      540     150

   3     2943     2160     420

   4    10624     6000     900

   5    29375    13500    1650

   6    68256    26460    2730

   7   140287    47040    4200

   8   263168    77760    6120

   9   459999   121500    8550

  10   760000   181500   11550

  11  1199231   261360   15180

  12  1821312   365040   19500

  ...

MATHEMATICA

(* Define the R[n, k] := A302971(m, j)/A304042(m, j) *)

R[n_, k_] := 0

R[n_, k_] := (2 k + 1)*Binomial[2 k, k]*

   Sum[R[n, j]*Binomial[j, 2 k + 1]*(-1)^(j - 1)/(j - k)*

   BernoulliB[2 j - 2 k], {j, 2 k + 1, n}] /; 2 k + 1 <= n

R[n_, k_] := (2 n + 1)*Binomial[2 n, n] /; k == n;

(* Define the U(m, l, t) coefficients *)

U[m_, l_, t_] := (-1)^m Sum[Sum[Binomial[j, t] R[m, j] k^(2 j - t) (-1)^j, {j, t, m}], {k, 1, l}];

(* Define the value of the variable 'm', should be m = 2 for A316349 *)

m = 2;

(* Print first 10 rows of U(m, l, t) coefficients over l: 1 <= l <= 10 *)

Column[Table[U[m, l, t], {l, 1, 10}, {t, 0, m}]]

CROSSREFS

The case m=1 is A320047.

The case m=3 is A316387.

Column k=0 is A316457.

Column k=1 is A316458.

Column k=2 is A316459.

Cf. A302971, A304042, A287326, A300656, A300785.

Sequence in context: A153636 A109840 A108293 * A063339 A228541 A115833

Adjacent sequences:  A316346 A316347 A316348 * A316350 A316351 A316352

KEYWORD

nonn,tabf

AUTHOR

Kolosov Petro, Jun 29 2018

EXTENSIONS

Edited by Max Alekseyev, Sep 06 2018

STATUS

approved

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Last modified February 23 08:18 EST 2019. Contains 320420 sequences. (Running on oeis4.)