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A026009 Triangular array T read by rows: T(n,0) = 1 for n >= 0; T(1,1) = 1; and for n >= 2, T(n,k) = T(n-1,k-1) + T(n-1,k) for k = 1,2,...,[(n+1)/2]; T(n,n/2 + 1) = T(n-1,n/2) if n is even. 19
1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 4, 6, 3, 1, 5, 10, 9, 1, 6, 15, 19, 9, 1, 7, 21, 34, 28, 1, 8, 28, 55, 62, 28, 1, 9, 36, 83, 117, 90, 1, 10, 45, 119, 200, 207, 90, 1, 11, 55, 164, 319, 407, 297, 1, 12, 66, 219, 483, 726, 704, 297, 1, 13, 78, 285, 702, 1209, 1430, 1001, 1, 14, 91, 363, 987, 1911, 2639, 2431, 1001 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

LINKS

G. C. Greubel, Rows n = 0..100 of the triangle, flattened

FORMULA

T(n, k) = binomial(n, k) - binomial(n, k-3). - Darko Marinov (marinov(AT)lcs.mit.edu), May 17 2001

Sum_{k=0..floor((n+2)/2)} T(n, k) = A026010(n). - G. C. Greubel, Mar 18 2021

EXAMPLE

From Jonathon Kirkpatrick, Jul 01 2016: (Start)

Triangle begins:

1;

1, 1;

1, 2, 1;

1, 3, 3;

1, 4, 6, 3;

1, 5, 10, 9;

1, 6, 15, 19, 9;

1, 7, 21, 34, 28;

1, 8, 28, 55, 62, 28;

1, 9, 36, 83, 117, 90;

1, 10, 45, 119, 200, 207, 90;

1, 11, 55, 164, 319, 407, 297;

1, 12, 66, 219, 483, 726, 704, 297;

1, 13, 78, 285, 702, 1209, 1430, 1001;

... (End)

MATHEMATICA

T[n_, k_]:= Binomial[n, k] - Binomial[n, k-3];

Join[{1}, Table[T[n, k], {n, 14}, {k, 0, Floor[(n+2)/2]}]//Flatten] (* G. C. Greubel, Mar 18 2021 *)

PROG

(Sage) [1]+flatten([[binomial(n, k) - binomial(n, k-3) for k in (0..(n+2)//2)] for n in (1..15)]) # G. C. Greubel, Mar 18 2021

(Magma) [1] cat [Binomial(n, k) - Binomial(n, k-3): k in [0..Floor((n+2)/2)], n in [1..15]]; // G. C. Greubel, Mar 18 2021

CROSSREFS

Diagonals of this sequence: A000217, A000245, A026012, A026013, A026014, A026015, A026016, A026017, A026018, A026019, A026020, A026021.

Sums involving this sequence: A026010, A027287, A027288, A027289, A027290, A027291, A027292.

Sequence in context: A299208 A334187 A082870 * A137171 A010356 A100640

Adjacent sequences: A026006 A026007 A026008 * A026010 A026011 A026012

KEYWORD

nonn,tabf,easy

AUTHOR

Clark Kimberling

STATUS

approved

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Last modified February 3 01:46 EST 2023. Contains 360024 sequences. (Running on oeis4.)