A101853 a(n) = n*(20 + 15*n + n^2)/6.

6, 18, 37, 64, 100, 146, 203, 272, 354, 450, 561, 688, 832, 994, 1175, 1376, 1598, 1842, 2109, 2400, 2716, 3058, 3427, 3824, 4250, 4706, 5193, 5712, 6264, 6850, 7471, 8128, 8822, 9554, 10325, 11136, 11988, 12882, 13819, 14800

Offset

1,1

Comments

4th partial summation within series as series accumulate n times from an initial sequence of Euler Triangle's row 3: 1,4,1. The 1,4,1 is the left column, A101101 the second column, A008458 the third, A003215 the fourth column etc of the array in the example. a(n) is the 4th row.

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

C. Rossiter, Depictions, Explorations and Formulas of the Euler/Pascal Cube.

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Formula

G.f.: x*(6 - 6*x + x^2)/(x - 1)^4. - R. J. Mathar, Dec 06 2011

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Vincenzo Librandi, Jun 26 2012

E.g.f.: exp(x)*x*(36 + 18*x + x^2)/6. - Stefano Spezia, Oct 14 2022

Example

Left column the third row of A008292, and subsequent columns defined as partial sums along their preceding neighbors:
1 1  1   1   1    1    1     1     1     1     1
4 5  6   7   8    9   10    11    12    13    14
1 6 12  19  27   36   46    57    69    82    96  A051936
0 6 18  37  64  100  146   203   272   354   450  A101853
0 6 24  61 125  225  371   574   846  1200  1650  A101854
0 6 30  91 216  441  812  1386  2232  3432  5082  A101855
0 6 36 127 343  784 1596  2982  5214  8646 13728
0 6 42 169 512 1296 2892  5874 11088 19734 33462
0 6 48 217 729 2025 4917 10791 21879 41613 75075
...

Mathematica

LinearRecurrence[{4, -6, 4, -1}, {6, 18, 37, 64}, 40] (* or *) CoefficientList[Series[(6-6*x+x^2)/(x-1)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Jun 26 2012 *)

Prog

(Magma)  I:=[6, 18, 37, 64]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..40]]; // Vincenzo Librandi, Jun 26 2012
(PARI) a(n)=n*(20+15*n+n^2)/6 \\ Charles R Greathouse IV, Oct 16 2015

Crossrefs

Row n=3 of A255961.

Cf. A003215, A008292, A008458, A101101.

Sequence in context: A202366 A185223 A299272 * A132432 A005899 A261652

Adjacent sequences: A101850 A101851 A101852 * A101854 A101855 A101856

Keyword

easy,nonn

Author

Cecilia Rossiter (cecilia(AT)noticingnumbers.net), Dec 18 2004

Status

approved