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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 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} binomial(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 May 30 08:04 EDT 2020. Contains 334712 sequences. (Running on oeis4.)