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A134869 Row sums of triangle A134868. 4
1, 4, 7, 11, 16, 22, 29, 37, 46, 56, 67, 79, 92, 106, 121, 137, 154, 172, 191, 211, 232, 254, 277, 301, 326, 352, 379, 407, 436, 466, 497, 529, 562, 596, 631, 667, 704, 742, 781, 821, 862, 904, 947, 991, 1036, 1082, 1129, 1177, 1226, 1276, 1327, 1379, 1432 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Where records occur in A182703. - Omar E. Pol, Feb 14 2012

Consider quadratic polynomials x^2+cx+d. Then a(n) is the number of these polynomials with 0 <= c < n, 0 <= d < n where no polynomial can be horizontally translated into another. For example, a(3) = 7, the coefficients are as follows: (c, d) = {(0, 0), (0, 1), (0, 2), (1, 0), (1, 1), (1, 2), (2, 0)}. Two polynomials are excluded, namely x^2+2x+1 = (x+1)^2+0(x+1)+0, and x^2+2x+2 = (x+1)^2+0(x+1)+1. - Griffin N. Macris, Jul 19 2016

a(n) gives the number of regions into which the square [0,1]x[0,1] is divided by the Bernstein polynomials of degree n. - Franck Maminirina Ramaharo, Feb 28 2018

LINKS

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

FORMULA

a(n) = 1, then for n>1, a(n) = T(n) + 1, where A000217 = (1, 3, 6, 10, 15, ...).

Binomial transform of [1, 3, 0, 1, -1, 1, -1, 1, ...].

From R. J. Mathar, Oct 27 2008: (Start)

G.f.: x(1+x-2x^2+x^3)/(1-x)^3.

a(n) = 1 + A000217(n) = A000124(n), n > 1. (End)

EXAMPLE

a(4) = 11 = sum of row 4 terms of triangle A134868: (2, + 2 + 3 + 4).

a(4) = 11 = 1 + 10, where 10 = T(4).

a(4) = 11 = (1, 3, 3, 1) dot (1, 3, 0, 1) = (1 + 9 + 0 + 1).

MAPLE

a:=n->sum((stirling2(j+1, n)), j=1..n):seq(a(n), n=1..50); # Zerinvary Lajos, Apr 12 2008

MATHEMATICA

Table[(n^2 + n)/2 + Boole[n != 1], {n, 53}] (* or *)

Table[PolygonalNumber@ n + Boole[n != 1], {n, 53}] (* Version 10.4, or *)

Table[Sum[StirlingS2[k + 1, n], {k, n}], {n, 53}] (* or *)

Rest@ CoefficientList[Series[x (1 + x - 2 x^2 + x^3)/(1 - x)^3, {x, 0, 53}], x] (* Michael De Vlieger, Jul 19 2016 *)

PROG

(PARI) a(n)=if(n>1, n*(n+1)/2+1, 1) \\ Charles R Greathouse IV, Aug 05 2016

CROSSREFS

Cf. A000217, A134868.

Sequence in context: A269132 A310757 A231603 * A171377 A020683 A310758

Adjacent sequences:  A134866 A134867 A134868 * A134870 A134871 A134872

KEYWORD

nonn,easy

AUTHOR

Gary W. Adamson, Nov 14 2007

EXTENSIONS

More terms from R. J. Mathar, Oct 27 2008

STATUS

approved

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Last modified December 11 20:17 EST 2018. Contains 318051 sequences. (Running on oeis4.)