A317981 -id:A317981 - cp's OEIS frontend

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.

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

Offset: 1

Kolosov Petro, Jul 01 2018

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.

			            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
  ...

Max Alekseyev, Derivation of the general formula for U(m,n,k), MathOverflow, 2018.
Petro Kolosov, On the link between binomial theorem and discrete convolution, arXiv:1603.02468 [math.NT], 2016-2025.
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.
Petro Kolosov, History and overview of the polynomial P_b^m(x), 2024.
Petro Kolosov, An efficient method of spline approximation for power function, arXiv:2503.07618 [math.GM], 2025.

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.

(* 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}]]

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

Edited by Max Alekseyev, Sep 06 2018

A317982 Expansion of 14x(29 + 784x + 1974x^2 + 784x^3 + 29x^4) / (1 - x)^7.

Original entry on oeis.org

406, 13818, 115836, 545860, 1858290, 5124126, 12182968, 25945416, 50745870, 92745730, 160386996, 264896268, 420839146, 646725030, 965662320, 1406064016, 2002403718, 2796022026, 3835983340, 5179983060, 6895305186, 9059830318, 11763094056, 15107395800

Offset: 1

Colin Barker, Aug 13 2018

Seems to be the second column of A316387.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A316387, A317981, A317983, A317984.

LinearRecurrence[{7,-21,35,-35,21,-7,1},{406,13818,115836,545860,1858290,5124126,12182968},30] (* Harvey P. Dale, Nov 15 2022 *)

Vec(14*x*(29 + 784*x + 1974*x^2 + 784*x^3 + 29*x^4) / (1 - x)^7 + O(x^40))

a(n) = 70*n^6 + 210*n^5 + 175*n^4 - 42*n^2 - 7*n

G.f.: 14*x*(29 + 784*x + 1974*x^2 + 784*x^3 + 29*x^4) / (1 - x)^7.
a(n) = 70*n^6 + 210*n^5 + 175*n^4 - 42*n^2 - 7*n.
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7) for n>7.

A317983 Expansion of 420x(1 + x)(1 + 10x + x^2) / (1 - x)^6.

Original entry on oeis.org

420, 7140, 41160, 148680, 411180, 955500, 1963920, 3684240, 6439860, 10639860, 16789080, 25498200, 37493820, 53628540, 74891040, 102416160, 137494980, 181584900, 236319720, 303519720, 385201740, 483589260, 601122480, 740468400, 904530900, 1096460820

Offset: 1

Colin Barker, Aug 13 2018

Seems to be the negative of the third column of A316387.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A000538, A316387, A317981, A317982, A317984.

Vec(420*x*(1 + x)*(1 + 10*x + x^2) / (1 - x)^6 + O(x^40))

a(n) = 84*n^5 + 210*n^4 + 140*n^3 - 14*n

G.f.: 420*x*(1 + x)*(1 + 10*x + x^2) / (1 - x)^6.
a(n) = 420 * A000538(n).
a(n) = 84*n^5 + 210*n^4 + 140*n^3 - 14*n.
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) for n>6.

A317984 Expansion of 140x(1 + 4*x + x^2) / (1 - x)^5.

Original entry on oeis.org

140, 1260, 5040, 14000, 31500, 61740, 109760, 181440, 283500, 423500, 609840, 851760, 1159340, 1543500, 2016000, 2589440, 3277260, 4093740, 5054000, 6174000, 7470540, 8961260, 10664640, 12600000, 14787500, 17248140, 20003760, 23077040, 26491500, 30271500

Offset: 1

Colin Barker, Aug 13 2018

Seems to be the fourth column of A316387.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A316387, A317981, A317982, A317983.

Vec(140*x*(1 + 4*x + x^2) / (1 - x)^5 + O(x^40))

PARI
```
a(n) = 35*n^4 + 70*n^3 + 35*n^2
```

G.f.: 140*x*(1 + 4*x + x^2) / (1 - x)^5.
a(n) = 35*n^4 + 70*n^3 + 35*n^2.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n>5.

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A317982 Expansion of 14*x*(29 + 784*x + 1974*x^2 + 784*x^3 + 29*x^4) / (1 - x)^7.

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A317983 Expansion of 420*x*(1 + x)*(1 + 10*x + x^2) / (1 - x)^6.

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A317984 Expansion of 140*x*(1 + 4*x + x^2) / (1 - x)^5.

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A317982 Expansion of 14x(29 + 784x + 1974x^2 + 784x^3 + 29x^4) / (1 - x)^7.

A317983 Expansion of 420x(1 + x)(1 + 10x + x^2) / (1 - x)^6.

A317984 Expansion of 140x(1 + 4*x + x^2) / (1 - x)^5.