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A014473 Pascal's triangle - 1. 14
0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 3, 5, 3, 0, 0, 4, 9, 9, 4, 0, 0, 5, 14, 19, 14, 5, 0, 0, 6, 20, 34, 34, 20, 6, 0, 0, 7, 27, 55, 69, 55, 27, 7, 0, 0, 8, 35, 83, 125, 125, 83, 35, 8, 0, 0, 9, 44, 119, 209, 251, 209, 119, 44, 9, 0, 0, 10, 54, 164, 329, 461, 461, 329, 164, 54, 10, 0, 0 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,8

COMMENTS

T(n,k) = A109128(n,k) - A007318(n,k), 0 <= k <= n. - Reinhard Zumkeller, Apr 10 2012

Indexed as a square array A(n,k): If X is an (n+k)-set and Y a fixed k-subset of X then A(n,k) is equal to the number of n-subsets of X intersecting Y. - Peter Luschny, Apr 20 2012

LINKS

Reinhard Zumkeller, Rows n=0..100 of triangle, flattened

Milan Janjic, Two Enumerative Functions

FORMULA

G.f.: x^2*y/((1 - x)*(1 - x*y)*(1 - x*(1 + y))). - Ralf Stephan, Jan 24 2005

T(n, k) = T(n-1, k-1) + T(n-1, k) + 1, 0 < k < n with T(n, 0) = T(n, n) = 0. - Reinhard Zumkeller, Jul 18 2015

If seen as a square array read by antidiagonals the generating function of row n is: G(n) = (x - 1)^(-n - 1) + (-1)^(n + 1)/(x*(x - 1)). - Peter Luschny, Feb 13 2019

EXAMPLE

Triangle begins:

   0;

   0, 0;

   0, 1,  0;

   0, 2,  2,  0;

   0, 3,  5,  3,  0;

   0, 4,  9,  9,  4,  0;

   0, 5, 14, 19, 14,  5, 0;

   0, 6, 20, 34, 34, 20, 6, 0;

   ...

Seen as a square array read by antidiagonals:

    [0] 0, 0,  0,  0,   0,   0,   0,    0,    0,    0,    0,     0, ... A000004

    [1] 0, 1,  2,  3,   4,   5,   6,    7,    8,    9,   10,    11, ... A001477

    [2] 0, 2,  5,  9,  14,  20,  27,   35,   44,   54,   65,    77, ... A000096

    [3] 0, 3,  9, 19,  34,  55,  83,  119,  164,  219,  285,   363, ... A062748

    [4] 0, 4, 14, 34,  69, 125, 209,  329,  494,  714, 1000,  1364, ... A063258

    [5] 0, 5, 20, 55, 125, 251, 461,  791, 1286, 2001, 3002,  4367, ... A062988

    [6] 0, 6, 27, 83, 209, 461, 923, 1715, 3002, 5004, 8007, 12375, ... A124089

MAPLE

with(combstruct): for n from 0 to 11 do seq(-1+count(Combination(n), size=m), m = 0 .. n) od; # Zerinvary Lajos, Apr 09 2008

# The rows of the square array:

Arow := proc(n, len) local gf, ser;

gf := (x - 1)^(-n - 1) + (-1)^(n + 1)/(x*(x - 1));

ser := series(gf, x, len+2): seq((-1)^(n+1)*coeff(ser, x, j), j=0..len) end:

for n from 0 to 9 do lprint([n], Arow(n, 12)) od; # Peter Luschny, Feb 13 2019

PROG

(Haskell)

a014473 n k = a014473_tabl !! n !! k

a014473_row n = a014473_tabl !! n

a014473_tabl = map (map (subtract 1)) a007318_tabl

-- Reinhard Zumkeller, Apr 10 2012

CROSSREFS

T(n,k) = A007318(n,k) - 1. A007318(n,k) + 1 = A323211(n,k).

Triangle without zeros: A014430. Row sums are in A000295.

Columns include A000096, A062748, A063258, A062988.

Diagonals of the square array A(n, n+d): A030662 (d=0), A010763 (d=1), A322938 (d=2).

Sequence in context: A004247 A271916 A327031 * A226545 A271917 A185651

Adjacent sequences:  A014470 A014471 A014472 * A014474 A014475 A014476

KEYWORD

nonn,tabl,easy

AUTHOR

N. J. A. Sloane.

EXTENSIONS

More terms from Erich Friedman.

STATUS

approved

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Last modified September 18 20:13 EDT 2019. Contains 327181 sequences. (Running on oeis4.)