A316387 -id:A316387 - cp's OEIS frontend

A300785 Triangle read by rows: T(n,k) = 140k^3(n-k)^3 - 14k(n-k) + 1; n >= 0, 0 <= k <= n.

1, 1, 1, 1, 127, 1, 1, 1093, 1093, 1, 1, 3739, 8905, 3739, 1, 1, 8905, 30157, 30157, 8905, 1, 1, 17431, 71569, 101935, 71569, 17431, 1, 1, 30157, 139861, 241753, 241753, 139861, 30157, 1, 1, 47923, 241753, 472291, 573217, 472291, 241753, 47923, 1, 1, 71569, 383965, 816229, 1119721, 1119721, 816229, 383965, 71569, 1

Offset: 0

Kolosov Petro, Mar 12 2018

From Kolosov Petro, Apr 12 2020: (Start)
Let A(m, r) = A302971(m, r) / A304042(m, r).
Let L(m, n, k) = Sum_{r=0..m} A(m, r) * k^r * (n - k)^r.
Then T(n, k) = L(3, n, k).
T(n, k) is symmetric: T(n, k) = T(n, n-k). (End)

			Triangle begins:
--------------------------------------------------------------------
k=   0      1       2       3       4       5       6      7     8
--------------------------------------------------------------------
n=0: 1;
n=1: 1,     1;
n=2: 1,   127,      1;
n=3: 1,  1093,   1093,      1;
n=4: 1,  3739,   8905,   3739,      1;
n=5: 1,  8905,  30157,  30157,   8905,      1;
n=6: 1, 17431,  71569, 101935,  71569,  17431,      1;
n=7: 1, 30157, 139861, 241753, 241753, 139861,  30157,     1;
n=8: 1, 47923, 241753, 472291, 573217, 472291, 241753, 47923,    1;

Muniru A Asiru, Rows n=0..100 of triangle, flattened.
Petro Kolosov, On the link between binomial theorem and discrete convolution, arXiv:1603.02468 [math.NT], 2016-2025.
Petro Kolosov, Polynomial identity involving binomial theorem and Faulhaber's formula, 2023.
Petro Kolosov, History and overview of the polynomial P_b^m(x), 2024.

Various cases of L(m, n, k): A287326 (m=1), A300656 (m=2), This sequence (m=3). See comments for L(m, n, k).
Row sums give A258806.
Cf. A000584, A287326, A007318, A077028, A294317, A068236, A300656, A302971, A304042, A001015, A094053, A258808, A024005, A316387.

T:=Flat(List([0..9], n->List([0..n], k->140*k^3*(n-k)^3 - 14*k*(n-k)+1))); # G. C. Greubel, Dec 14 2018

/* As triangle */ [[140*k^3*(n-k)^3-14*k*(n-k)+1: k in [0..n]]: n in [0..10]]; // Bruno Berselli, Mar 21 2018

T:=(n,k)->140*k^3*(n-k)^3-14*k*(n-k)+1: seq(seq(T(n,k),k=0..n),n=0..9); # Muniru A Asiru, Dec 14 2018

T[n_, k_] := 140*k^3*(n - k)^3 - 14*k*(n - k) + 1; Column[
Table[T[n, k], {n, 0, 10}, {k, 0, n}], Center] (* From Kolosov Petro, Apr 12 2020 *)

t(n, k) = 140*k^3*(n-k)^3-14*k*(n-k)+1
trianglerows(n) = for(x=0, n-1, for(y=0, x, print1(t(x, y), ", ")); print(""))
/* Print initial 9 rows of triangle as follows */ trianglerows(9)

[[140*k^3*(n-k)^3 - 14*k*(n-k)+1 for k in range(n+1)] for n in range(12)] # G. C. Greubel, Dec 14 2018

From Kolosov Petro, Apr 12 2020: (Start)
T(n, k) = 140*k^3*(n-k)^3 - 14*k*(n-k) + 1.
T(n, k) = 140*A094053(n, k)^3 + 0*A094053(n, k)^2 - 14*A094053(n, k)^1 + 1.
T(n+3, k) = 4*T(n+2, k) - 6*T(n+1, k) + 4*T(n, k) - T(n-1, k), for n >= k.
Sum_{k=1..n} T(n, k) = A001015(n).
Sum_{k=0..n} T(n, k) = A258806(n).
Sum_{k=0..n-1} T(n, k) = A001015(n).
Sum_{k=1..n-1} T(n, k) = A258808(n).
Sum_{k=1..n-1} T(n, k) = -A024005(n).
Sum_{k=1..r} T(n, k) = -A316387(3, r, 0)*n^0 + A316387(3, r, 1)*n^1 - A316387(3, r, 2)*n^2 + A316387(3, r, 3)*n^3. (End)
G.f.: (1 + 127*x^6*y^3 - 3*x*(1 + y) + 585*x^5*y^2*(1 + y) + 129*x^4*y*(1 + 17*y + y^2) + 3*x^2*(1 + 45*y + y^2) - x^3*(1 - 579*y - 579*y^2 + y^3))/((1 - x)^4*(1 - x*y)^4). - Stefano Spezia, Sep 14 2024

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.

Original entry on oeis.org

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

Offset: 1

Kolosov Petro, Jun 29 2018

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.

			       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
  ...

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=3 is A316387.
Column k=0 is A316457.
Column k=1 is A316458.
Column k=2 is A316459.
Cf. A302971, A304042, A287326, A300656, A300785.

(* 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}]]

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.

Edited by Max Alekseyev, Sep 06 2018

A317981 Expansion of x(125 + 8028x + 42237x^2 + 42272x^3 + 8007x^4 + 132x^5 - x^6) / (1 - x)^8.

Original entry on oeis.org

125, 9028, 110961, 684176, 2871325, 9402660, 25872833, 62572096, 136972701, 276971300, 524988145, 943023888, 1618774781, 2672907076, 4267591425, 6616398080, 9995653693, 14757360516, 21343778801, 30303773200, 42311023965, 58184203748, 78909220801

Offset: 1

Colin Barker, Aug 13 2018

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

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Cf. A316387, A317982, A317983, A317984.

LinearRecurrence[{8,-28,56,-70,56,-28,8,-1},{125,9028,110961,684176,2871325,9402660,25872833,62572096},30] (* Harvey P. Dale, Dec 29 2024 *)

Vec(x*(125 + 8028*x + 42237*x^2 + 42272*x^3 + 8007*x^4 + 132*x^5 - x^6) / (1 - x)^8 + O(x^40))

a(n) = 20*n^7 + 70*n^6 + 70*n^5 - 28*n^3 - 7*n^2

G.f.: x*(125 + 8028*x + 42237*x^2 + 42272*x^3 + 8007*x^4 + 132*x^5 - x^6) / (1 - x)^8.
a(n) = 20*n^7 + 70*n^6 + 70*n^5 - 28*n^3 - 7*n^2.
a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8) for n>8.

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.

A320047 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 2-column table read by rows, where n-th row lists coefficients U(1,n,k) for k = 0, 1 and n >= 1.

Original entry on oeis.org

5, 6, 28, 18, 81, 36, 176, 60, 325, 90, 540, 126, 833, 168, 1216, 216, 1701, 270, 2300, 330, 3025, 396, 3888, 468, 4901, 546, 6076, 630, 7425, 720, 8960, 816, 10693, 918, 12636, 1026, 14801, 1140, 17200, 1260, 19845, 1386, 22748, 1518, 25921, 1656

Offset: 1

Kolosov Petro, Oct 04 2018

For l=T, the identity takes the 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
   l      k=0     k=1
  ---   ------  ------
   1       5       6
   2      28      18
   3      81      36
   4     176      60
   5     325      90
   6     540     126
   7     833     168
   8    1216     216
   9    1701     270
  10    2300     330
  11    3025     396
  12    3888     468
  ...

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=2 is A316349.
The case m=3 is A316387.
Column k=0 is A275709.
Column k=1 is A028896.
Cf. A302971, A304042, A287326, A300656, A300785.

(* Define the R[n,k] := A302971(n,k)/A304042(n,k) *)
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' to be m = 1 for A320047 *)
m = 1;
(* 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(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.
Conjectures from Colin Barker, Aug 03 2019: (Start)
G.f.: x*(5 + 6*x + 8*x^2 - 6*x^3 - x^4) / ((1 - x)^4*(1 + x)^4).
a(n) = (4 - 4*(-1)^n - 3*(-5+(-1)^n)*n - 3*(-3+(-1)^n)*n^2 + (1+(-1)^(1+n))*n^3) / 8.
a(n) = 4*a(n-2) - 6*a(n-4) + 4*a(n-6) - a(n-8) for n>8.
(End)

cp's OEIS Frontend

A300785 Triangle read by rows: T(n,k) = 140*k^3*(n-k)^3 - 14*k*(n-k) + 1; n >= 0, 0 <= k <= n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A317981 Expansion of x*(125 + 8028*x + 42237*x^2 + 42272*x^3 + 8007*x^4 + 132*x^5 - x^6) / (1 - x)^8.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

PARI

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

PARI

Formula

Views

Author

Keywords

Comments

Examples

Links

A300785 Triangle read by rows: T(n,k) = 140k^3(n-k)^3 - 14k(n-k) + 1; n >= 0, 0 <= k <= n.

A317981 Expansion of x(125 + 8028x + 42237x^2 + 42272x^3 + 8007x^4 + 132x^5 - x^6) / (1 - x)^8.

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.