The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A316387 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 4-column table read by rows, where the n-th row lists coefficients U(3,n,k) for k = 0, 1, 2, 3; n >= 1. 7
 125, 406, 420, 140, 9028, 13818, 7140, 1260, 110961, 115836, 41160, 5040, 684176, 545860, 148680, 14000, 2871325, 1858290, 411180, 31500, 9402660, 5124126, 955500, 61740, 25872833, 12182968, 1963920, 109760, 62572096, 25945416, 3684240, 181440, 136972701, 50745870, 6439860, 283500, 276971300, 92745730, 10639860, 423500 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS For T=L, 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, Another Power Identity involving Binomial Theorem and Faulhaber's formula, arXiv:1603.02468 [math.NT], 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(3,n,0) = 20*n^7 + 70*n^6 + 70*n^5 - 28*n^3 - 7*n^2; U(3,n,1) = 70*n^6 + 210*n^5 + 175*n^4 - 42*n^2 - 7*n; U(3,n,2) = 84*n^5 + 210*n^4 + 140*n^3 - 14*n; U(3,n,3) = 35*n^4 + 70*n^3 + 35*n^2. - Max Alekseyev, Sep 06 2018 From Colin Barker, Jul 09 2018; corrected by Max Alekseyev, Sep 06 2018: (Start) G.f.: x*(125 + 406*x + 420*x^2 + 140*x^3 + 8028*x^4 + 10570*x^5 + 3780*x^6 + 140*x^7 + 42237*x^8 + 16660*x^9 - 4200*x^10 - 1120*x^11 + 42272*x^12 - 16660*x^13 - 4200*x^14 + 1120*x^15 + 8007*x^16 - 10570*x^17 + 3780*x^18 - 140*x^19 + 132*x^20 - 406*x^21 + 420*x^22 - 140*x^23 - x^24) / ((1 - x)^8*(1 + x)^8*(1 + x^2)^8). a(n) = 8*a(n-4) - 28*a(n-8) + 56*a(n-12) - 70*a(n-16) + 56*a(n-20) - 28*a(n-24) + 8*a(n-28) - a(n-32) for n>32. (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. - Kolosov Petro, Oct 04 2018 EXAMPLE column      column      column   column    L          k=0         k=1         k=2      k=3   --  ------------  ----------  ----------  -------    1           125         406         420      140    2          9028       13818        7140     1260    3        110961      115836       41160     5040    4        684176      545860      148680    14000    5       2871325     1858290      411180    31500    6       9402660     5124126      955500    61740    7      25872833    12182968     1963920   109760    8      62572096    25945416     3684240   181440    9     136972701    50745870     6439860   283500   10     276971300    92745730    10639860   423500   11     524988145   160386996    16789080   609840   12     943023888   264896268    25498200   851760   13    1618774781   420839146    37493820  1159340   14    2672907076   646725030    53628540  1543500   15    4267591425   965662320    74891040  2016000   16    6616398080  1406064016   102416160  2589440   17    9995653693  2002403718   137494980  3277260   18   14757360516  2796022026   181584900  4093740   19   21343778801  3835983340   236319720  5054000   20   30303773200  5179983060   303519720  6174000   ... 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' *) m = 3; (* Print first 10 rows of U(m, l, t) coefficients for 'm' defined above *) Column[Table[U[m, l, t], {l, 1, 10}, {t, 0, m}]] CROSSREFS The case m=1 is A320047. The case m=2 is A316349. Column k=0 is A317981. Column k=1 is A317982. Column k=2 is A317983. Column k=3 is A317984. Cf. A287326, A300656, A300785, A302971, A304042. Sequence in context: A252057 A045184 A059470 * A250900 A293040 A250136 Adjacent sequences:  A316384 A316385 A316386 * A316388 A316389 A316390 KEYWORD nonn,tabf AUTHOR Kolosov Petro, Jul 01 2018 EXTENSIONS Edited by Max Alekseyev, Sep 06 2018 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified May 25 13:19 EDT 2022. Contains 354071 sequences. (Running on oeis4.)