A272039 -id:A272039 - cp's OEIS frontend

A228219 Number of second differences of arrays of length 4 of numbers in 0..n.

15, 49, 103, 177, 271, 385, 519, 673, 847, 1041, 1255, 1489, 1743, 2017, 2311, 2625, 2959, 3313, 3687, 4081, 4495, 4929, 5383, 5857, 6351, 6865, 7399, 7953, 8527, 9121, 9735, 10369, 11023, 11697, 12391, 13105, 13839, 14593, 15367, 16161, 16975, 17809

Offset: 1

R. H. Hardin, Aug 16 2013

nonn

Row 2 of A228218.

			Some solutions for n=4:
..1....1....1....5....6....1....2...-5....8....4...-5....8....1....5...-7....0
..4....2...-3....1...-6....1....1...-1...-7....1....0...-4....5...-7....6....6

R. H. Hardin, Table of n, a(n) for n = 1..210

Cf. A228218, A272039.

Empirical: a(n) = 10*n^2 + 4*n + 1 = A272039(n).
Conjectures from Colin Barker, Mar 16 2018: (Start)
G.f.: x*(15 + 4*x + x^2) / (1 - x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>3.
(End)

A272124 a(n) = 12n^4 + 16n^3 + 10n^2 + 4n + 1.

Original entry on oeis.org

1, 43, 369, 1507, 4273, 9771, 19393, 34819, 58017, 91243, 137041, 198243, 277969, 379627, 506913, 663811, 854593, 1083819, 1356337, 1677283, 2052081, 2486443, 2986369, 3558147, 4208353, 4943851, 5771793, 6699619, 7735057, 8886123, 10161121, 11568643

Offset: 0

Vincenzo Librandi, Apr 21 2016

M. Beck, J. A. De Loera, M. Develin, J. Pfeifle and R. P. Stanley, Coefficients and roots of Ehrhart Polynomials, Contemp. Math. 374 (2005), 15-36, page 19.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A010684, A272039.

[12*n^4+16*n^3+10*n^2+4*n+1: n in [0..50]];

A272124:=n->(12*n^4 + 16*n^3 + 10*n^2 + 4*n + 1): seq(A272124(n), n=0..60); # Wesley Ivan Hurt, Apr 22 2016

LinearRecurrence[{5, -10, 10, -5, 1}, {1, 43, 369, 1507, 4273}, 50]
CoefficientList[Series[(1 + 38*x + 164*x^2 + 82*x^3 + 3*x^4)/(1 - x)^5, {x, 0, 30}], x] (* Wesley Ivan Hurt, Apr 22 2016 *)

vector(100, n, n--; 12*n^4+16*n^3+10*n^2+4*n+1) \\ Altug Alkan, Apr 22 2016

O.g.f.: (1+38*x+164*x^2+82*x^3+3*x^4)/(1-x)^5.
E.g.f.: (1+42*x+142*x^2+88*x^3+12*x^4)*exp(x).
a(n) = 5*a(n-1)-10*a(n-2)+10*a(n-3)-5*a(n-4)+a(n-5) for n>4.
a(n) mod 4 = a(n) mod 8 = A010684(n). - Wesley Ivan Hurt, Apr 22 2016

A272130 a(n) = 16n^3 + 10n^2 + 4*n + 1.

Original entry on oeis.org

1, 31, 177, 535, 1201, 2271, 3841, 6007, 8865, 12511, 17041, 22551, 29137, 36895, 45921, 56311, 68161, 81567, 96625, 113431, 132081, 152671, 175297, 200055, 227041, 256351, 288081, 322327, 359185, 398751, 441121, 486391, 534657, 586015, 640561, 698391

Offset: 0

Vincenzo Librandi, Apr 21 2016

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
M. Beck, J. A. De Loera, M. Develin, J. Pfeifle and R. P. Stanley, Coefficients and roots of Ehrhart Polynomials, Contemp. Math. 374 (2005), 15-36, page 19.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A144965, A158187, A272039.

Magma
```
[16*n^3+10*n^2+4*n+1: n in [0..50]];
```

A272130:=n->16*n^3+10*n^2+4*n+1: seq(A272130(n), n=0..50); # Wesley Ivan Hurt, Apr 22 2016

LinearRecurrence[{4,-6,4,-1},{1,31,177,535},50]
CoefficientList[Series[(1 + 27*x + 59*x^2 + 9*x^3)/(1 - x)^4, {x, 0, 30}], x] (* Wesley Ivan Hurt, Apr 22 2016 *)

vector(100, n, n--; 16*n^3+10*n^2+4*n+1) \\ Altug Alkan, Apr 22 2016

O.g.f.: (1+27*x+59*x^2+9*x^3)/(1-x)^4.
E.g.f.: (1+30*x+58*x^2+16*x^3)*exp(x).
a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4) for n>3.
a(n) = A158187(n) + A144965(n).

A338369 Triangle read by rows: T(n,k) = (Sum_{i=0..n-k}(1+ki)^3)/(Sum_{i=0..n-k} (1+ki)) for 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 6, 7, 1, 1, 10, 17, 13, 1, 1, 15, 31, 34, 21, 1, 1, 21, 49, 64, 57, 31, 1, 1, 28, 71, 103, 109, 86, 43, 1, 1, 36, 97, 151, 177, 166, 121, 57, 1, 1, 45, 127, 208, 261, 271, 235, 162, 73, 1, 1, 55, 161, 274, 361, 401, 385, 316, 209, 91, 1, 1, 66, 199, 349, 477, 556, 571, 519, 409, 262, 111, 1

Offset: 0

Werner Schulte, Nov 26 2020

Seen as a square array: (1) A(n,k) = T(n+k,k) = (k^2*n^2+k*(k+2)*n+2)/2 for n,k >= 0; (2) A(n,k) = A(n-1,k) + k*(1 + k*n) for k >= 0 and n > 0; (3) A(n,k) = A(n,k-1) + k*n*(n+1) - n*(n-1)/2 for n >= 0 and k > 0; (4) G.f. of row n >= 0: (2 + (n^2+3*n-4)*x + (n^2-n+2)*x^2) / (2*(1-x)^3).

			The triangle T(n,k) for 0 <= k <= n starts:
n \k :  0   1    2    3    4    5    6    7    8    9   10   11   12
====================================================================
   0 :  1
   1 :  1   1
   2 :  1   3    1
   3 :  1   6    7    1
   4 :  1  10   17   13    1
   5 :  1  15   31   34   21    1
   6 :  1  21   49   64   57   31    1
   7 :  1  28   71  103  109   86   43    1
   8 :  1  36   97  151  177  166  121   57    1
   9 :  1  45  127  208  261  271  235  162   73    1
  10 :  1  55  161  274  361  401  385  316  209   91    1
  11 :  1  66  199  349  477  556  571  519  409  262  111    1
  12 :  1  78  241  433  609  736  793  771  673  514  321  133    1
etc.

Cf. A000012 (column 0, main diagonal), A000217 (column 1), A056220 (column 2), A081271 (column 3), A118057 (column 4), A002061 (1st subdiagonal), A056109 (2nd subdiagonal), A085473 (3rd subdiagonal), A272039 (4th subdiagonal).

T[n_, k_] := Sum[(1 + k*i)^3, {i, 0, n - k}]/Sum[1 + k*i, {i, 0, n - k}]; Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Amiram Eldar, Nov 26 2020 *)

for(n=0,12,for(k=0,n,print1((k^2*(n-k)^2+k*(k+2)*(n-k)+2)/2,", "));print(" "))

T(n,k) = (k^2*(n-k)^2 + k*(k+2)*(n-k) + 2)/2 for 0 <= k <= n.
T(n,0) = T(n,n) = 1 for n >= 0.
T(n,k) = T(n-1,k-1) + k*(n-k)*(n-k+1) - (n-k)*(n-k-1)/2 for 0 < k <= n.
T(n,k) = T(n-1,k) + k * (1+k*(n-k)) for 0 <= k < n.
G.f. of column k >= 0: (1 + (k^2+k-2)*t + (1-k)*t^2) * t^k / (1-t)^3.
E.g.f.: exp(x+y)*(2 + (x^2 + 2*x - 2)*y + (x^2 - 4*x + 2)*y^2 - (2*x - 5)*y^3 + y^4)/2. - Stefano Spezia, Nov 27 2020

cp's OEIS Frontend

A228219 Number of second differences of arrays of length 4 of numbers in 0..n.

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Formula

A272124 a(n) = 12*n^4 + 16*n^3 + 10*n^2 + 4*n + 1.

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Author

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Magma

Maple

Mathematica

PARI

Formula

A272130 a(n) = 16*n^3 + 10*n^2 + 4*n + 1.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A338369 Triangle read by rows: T(n,k) = (Sum_{i=0..n-k}(1+k*i)^3)/(Sum_{i=0..n-k} (1+k*i)) for 0 <= k <= n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Formula

A272124 a(n) = 12n^4 + 16n^3 + 10n^2 + 4n + 1.

A272130 a(n) = 16n^3 + 10n^2 + 4*n + 1.

A338369 Triangle read by rows: T(n,k) = (Sum_{i=0..n-k}(1+ki)^3)/(Sum_{i=0..n-k} (1+ki)) for 0 <= k <= n.