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A101853 a(n) = n*(20+15*n+n^2)/6. 4
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 (list; graph; refs; listen; history; text; internal format)
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

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.

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

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Last modified February 20 12:12 EST 2018. Contains 299387 sequences. (Running on oeis4.)