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A126277 Triangle generated from Eulerian numbers. 5
1, 1, 2, 1, 3, 3, 1, 4, 7, 4, 1, 5, 11, 15, 5, 1, 6, 15, 26, 31, 6, 1, 7, 19, 37, 57, 63, 7, 1, 8, 23, 48, 83, 120, 127, 8, 1, 9, 27, 59, 109, 177, 247, 255, 9, 1, 10, 31, 70, 135, 234, 367, 502, 511, 10 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

N-th diagonal starting from the right = binomial transform of [1, N, q, q, q...) where q = 2*N - 2. Given the infinite set of triangles "T" composed of partial column sums of the polygonal numbers, the N-th diagonal starting from the right = row sums of triangle "T": (T=3 = A104712; T=4 = A125165; T=5 = A125232; T=6 = A125233; T=7 = A125234, T=8 = A125235; and so on). For example, 3rd diagonal from the right = the offset Eulerian numbers, (1, 4, 11, 26, 57, 120,...) = row sums of Triangle A104712 having partial column sums of the triangular numbers: 1; 3, 1; 6, 4, 1; 10, 10, 5, 1; 15, 20, 15, 6, 1; .. Row sums = A124671: (1, 3, 7, 16, 37, 85, 191,...)

LINKS

G. C. Greubel, Rows n=1..100 of triangle, flattened

FORMULA

Given right border = (1,2,3...), T(n,k) = A000295(k) + T(n-1,k); where A000295 = the Eulerian numbers starting (0, 1, 4, 11, 26, 57,...)

EXAMPLE

First few rows of the triangle are:

1;

1, 2;

1, 3, 3;

1, 4, 7, 4;

1, 5, 11, 15, 5;

1, 6, 15, 26, 31, 6;

1, 7, 19, 37, 57, 63, 7;

1, 8, 23, 48, 83, 120, 127, 8;

1, 9, 27, 59, 109, 177, 247, 255, 9;

1, 10, 31, 70, 135, 234, 367, 502, 511, 10;

...

T(7,4) = 37 = A000295(4) + T(6,4) = 11 + 26.

MATHEMATICA

T[n_, 1]:=1; T[n_, n_]:=n; T[n_, k_]:= T[n-1, k] + 2^k - k - 1; Table[T[n, k], {n, 1, 15}, {k, 1, n}]//Flatten (* G. C. Greubel, Oct 23 2018 *)

PROG

(PARI) {T(n, k) = if(k==1, 1, if(k==n, n, 2^k - k - 1 + T(n-1, k)))};

for(n=1, 10, for(k=1, n, print1(T(n, k), ", "))) \\ G. C. Greubel, Oct 23 2018

CROSSREFS

Cf. A126268, A000295, A104712, A125165, A104712, A125232 - A125235.

Sequence in context: A159881 A319539 A098546 * A253273 A055129 A133804

Adjacent sequences:  A126274 A126275 A126276 * A126278 A126279 A126280

KEYWORD

nonn,tabl

AUTHOR

_Gary w. Adamson_, Dec 23 2006

STATUS

approved

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Last modified September 23 15:58 EDT 2020. Contains 337310 sequences. (Running on oeis4.)