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A030528 Triangle read by rows: a(n,k) = binomial(k,n-k). 41
1, 1, 1, 0, 2, 1, 0, 1, 3, 1, 0, 0, 3, 4, 1, 0, 0, 1, 6, 5, 1, 0, 0, 0, 4, 10, 6, 1, 0, 0, 0, 1, 10, 15, 7, 1, 0, 0, 0, 0, 5, 20, 21, 8, 1, 0, 0, 0, 0, 1, 15, 35, 28, 9, 1, 0, 0, 0, 0, 0, 6, 35, 56, 36, 10, 1, 0, 0, 0, 0, 0, 1, 21, 70, 84, 45, 11, 1, 0, 0, 0, 0, 0, 0, 7, 56, 126, 120, 55, 12, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

A convolution triangle of numbers obtained from A019590.

a(n,m) := s1(-1; n,m), a member of a sequence of triangles including s1(0; n,m)= A023531(n,m) (unit matrix) and s1(2; n,m)= A007318(n-1,m-1) (Pascal's triangle).

The signed triangular matrix a(n,m)*(-1)^(n-m) is the inverse matrix of the triangular Catalan convolution matrix A033184(n+1,m+1), n >= m >= 0, with A033184(n,m) := 0 if n<m.

Riordan array (1+x, x(1+x)). The signed triangle is the Riordan array (1-x,x(1-x)), inverse to (c(x),xc(x)) with c(x) g.f. for A000108. - Paul Barry, Feb 02 2005

Also, a(n,k)=number of compositions of n into k parts of 1's and 2's. Example: a(6,4)=6 because we have 2211, 2121, 2112, 1221, 1212 and 1122. - Emeric Deutsch, Apr 05 2005

Subtriangle of A026729. - Philippe Deléham, Aug 31 2006

a(n,k) is the number of length n-1 binary sequences having no two consecutive 0's with exactly k-1 1's. Example: a(6,4)=6 because we have 01011, 01101, 01110, 10101, 10110, 11010. - Geoffrey Critzer, Jul 22 2013

Mirrored, shifted Fibonacci polynomials of A011973. The polynomials (illustrated below) of this entry have the property that p(n,t) = t * [p(n-1,t) + p(n-2,t)]. The additive properties of Pascal's triangle (A007318) are reflected in those of these polynomials, as can be seen in the Example Section below and also when the o.g.f. G(x,t) below is expanded as the series x*(1+x) + t * [x*(1+x)]^2 + t^2 * [x*(1+x)]^3 + ... . See also A053122 for a relation to Cartan matrices. - Tom Copeland, Nov 04 2014

Rows of this entry appear as columns of an array for an infinitesimal generator presented in the Copeland link. - Tom Copeland, Dec 23 2015

For n >= 2, the n-th row is also the coefficients of the vertex cover polynomial of the (n-1)-path graph P_{n-1}. - Eric W. Weisstein, Apr 10 2017

LINKS

Indranil Ghosh, Rows 1..125 of triangle, flattened

T. Copeland, Generators, Inversion, and Matrix, Binomial, and Integral Transforms, Addendum to Elliptic Lie Triad

W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

D. Merlini, R. Sprugnoli and M. C. Verri, An algebra for proper generating trees, Colloquium on Mathematics and Computer Science, Versailles, September 2000.

D. Merlini, R. Sprugnoli and M. C. Verri, An algebra for proper generating trees, Mathematics and Computer Science, Part of the series Trends in Mathematics pp 127-139.

Eric Weisstein's World of Mathematics, Path Graph

Eric Weisstein's World of Mathematics, Vertex Cover Polynomial

FORMULA

a(n, m) = 2*(2*m-n+1)*a(n-1, m)/n + m*a(n-1, m-1)/n, n >= m >= 1; a(n, m) := 0, n<m; a(n, 0) := 0; a(1, 1)=1.

G.f. for m-th column: (x*(1+x))^m.

As a number triangle with offset 0, this is T(n, k) = Sum_{k=0..n} (-1)^(n+i)binomial(n, i)binomial(i+k+1, 2k+1). The diagonal sums give the Padovan sequence A000931(n+5). Inverse binomial transform of A078812 (product of lower triangular matrices). - Paul Barry, Jun 21 2004

G.f.: (1 + x)/(1 - y*x - y*x^2). - Geoffrey Critzer, Jul 22 2013

From Tom Copeland, Nov 04 2014: (Start)

O.g.f: G(x,t) = x*(1+x) / [1 - t*x*(1+x)] = -P[Cinv(-x),t], where P(x,t)= x / (1 + t*x) and Cinv(x)= x*(1-x) are the compositional inverses in x of Pinv(x,t) = -P(-x,t) = x / (1 - t*x) and C(x) = [1-sqrt(1-4*x)]/2, an o.g.f. for the shifted Catalan numbers A000108.

Therefore, Ginv(x,t) = -C[Pinv(-x,t)] = {-1 + sqrt[1 + 4*x/(1+t*x)]}/2, which is  -A124644(-x,t).

This places this array in a family of arrays related by composition of P and C and their inverses and interpolation by t, such as A091867 and A104597, and associated to the Catalan, Motzkin, Fine, and Fibonacci numbers. Cf. A104597 (polynomials shifted in t) A125145, A146559, A057078, A000045, A155020, A125145, A039717, A001792, A057862, A011973, A115139. (End)

EXAMPLE

Triangle starts:

[ 1]  1

[ 2]  1   1

[ 3]  0   2   1

[ 4]  0   1   3   1

[ 5]  0   0   3   4   1

[ 6]  0   0   1   6   5   1

[ 7]  0   0   0   4  10   6   1

[ 8]  0   0   0   1  10  15   7   1

[ 9]  0   0   0   0   5  20  21   8   1

[10]  0   0   0   0   1  15  35  28   9   1

[11]  0   0   0   0   0   6  35  56  36  10   1

[12]  0   0   0   0   0   1  21  70  84  45  11   1

[13]  0   0   0   0   0   0   7  56 126 120  55  12   1

...

For quick comparison to other polynomials:

(From Tom Copeland, Nov 04 2014)

p(1,t) = 1

p(2,t) = 1 + 1 t

p(3,t) = 0 + 2 t + 1 t^2

p(4,t) = 0 + 1 t + 3 t^2 + 1 t^3

p(5,t) = 0 + 0   + 3 t^2 + 4 t^3 +  1 t^4

p(6,t) = 0 + 0   + 1 t^2 + 6 t^3 +  5 t^4 +  1 t^5

p(7,t) = 0 + 0   + 0     + 4 t^3 + 10 t^4 +  6 t^5 + 1 t^6

p(8,t) = 0 + 0   + 0     + 1 t^3 + 10 t^4 + 15 t^5 + 7 t^6 + 1 t^7

...

Reading along columns gives rows for Pascal's triangle.

MAPLE

for n from 1 to 12 do seq(binomial(k, n-k), k=1..n) od; # yields sequence in triangular form - Emeric Deutsch, Apr 05 2005

MATHEMATICA

nn=10; CoefficientList[Series[(1+x)/(1-y x - y x^2), {x, 0, nn}], {x, y}]//Grid (* Geoffrey Critzer, Jul 22 2013 *)

Table[Binomial[k, n - k], {n, 13}, {k, n}] // Flatten (* Michael De Vlieger, Dec 23 2015 *)

CoefficientList[Table[x^(n/2 - 1) Fibonacci[n + 1, Sqrt[x]], {n, 10}],

   x] // Flatten (* Eric W. Weisstein, Apr 10, 2017 *)

PROG

(MAGMA) /* As triangle */ [[Binomial(k, n-k): k in [1..n]]: n in [1.. 15]]; // Vincenzo Librandi, Nov 05 2014

CROSSREFS

Row sums A000045(n+1) (Fibonacci). a(n, 1)= A019590(n) (Fermat's last theorem). Cf. A049403.

Cf. A104597, A146559, A146559, A155020, A125145, A000045, A057078, A039717, A001792, A057862, A011973, A115139.

Sequence in context: A131185 A286354 A052249 * A077227 A089263 A156135

Adjacent sequences:  A030525 A030526 A030527 * A030529 A030530 A030531

KEYWORD

easy,nonn,tabl

AUTHOR

Wolfdieter Lang

EXTENSIONS

More terms from Emeric Deutsch, Apr 05 2005

STATUS

approved

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Last modified May 23 18:42 EDT 2017. Contains 286926 sequences.