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A104967 Matrix inverse of triangle A104219, read by rows, where A104219(n,k) equals the number of Schroeder paths of length 2n having k peaks at height 1. 6
1, -1, 1, -1, -2, 1, -1, -1, -3, 1, -1, 0, 0, -4, 1, -1, 1, 2, 2, -5, 1, -1, 2, 3, 4, 5, -6, 1, -1, 3, 3, 3, 5, 9, -7, 1, -1, 4, 2, 0, 0, 4, 14, -8, 1, -1, 5, 0, -4, -6, -6, 0, 20, -9, 1, -1, 6, -3, -8, -10, -12, -14, -8, 27, -10, 1, -1, 7, -7, -11, -10, -10, -14, -22, -21, 35, -11, 1, -1, 8, -12, -12, -5, 0, 0, -8, -27, -40, 44, -12, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Row sums equal A090132 with odd-indexed terms negated. Absolute row sums form A104968. Row sums of squared terms gives A104969.

Riordan array ((1-2*x)/(1-x), x(1-2*x)/(1-x)). - Philippe Deléham, Dec 05 2015

LINKS

Paul D. Hanna, Table of n, a(n) for n = 0..1080

FORMULA

G.f.: A(x, y) = (1-2*x)/(1-x - x*y*(1-2*x)).

Sum_{k=0..n} T(n, k) = (-1)^n*A090132(n).

Sum_{k=0..n} abs(T(n, k)) = A104968(n).

Sum_{k=0..n} T(n, k)^2 = A104969(n).

T(n,k) = Sum_{i=0..n-k} (-2)^i*binomial(k+1,i)*binomial(n-i,k). - Vladimir Kruchinin, Nov 02 2011

Sum_{k=0..floor(n/2)} T(n-k, k) = A078011(n+2). - G. C. Greubel, Jun 09 2021

EXAMPLE

Triangle begins:

1;

-1, 1;

-1, -2, 1;

-1, -1, -3, 1;

-1, 0, 0, -4, 1;

-1, 1, 2, 2, -5, 1;

-1, 2, 3, 4, 5, -6, 1;

-1, 3, 3, 3, 5, 9, -7, 1;

-1, 4, 2, 0, 0, 4, 14, -8, 1;

-1, 5, 0, -4, -6, -6, 0, 20, -9, 1; ...

MAPLE

A104967:= (n, k)-> add( (-2)^j*binomial(k+1, j)*binomial(n-j, k), j=0..n-k);

seq(seq( A104967(n, k), k=0..n), n=0..12); # G. C. Greubel, Jun 09 2021

MATHEMATICA

T[n_, k_]:= T[n, k]= Which[k==n, 1, k==0, 0, True, T[n-1, k-1] - Sum[T[n-i, k-1], {i, 2, n-k+1}]];

Table[T[n, k], {n, 13}, {k, n}]//Flatten (* Jean-François Alcover, Jun 11 2019, after Peter Luschny *)

PROG

(PARI) {T(n, k)=local(X=x+x*O(x^n), Y=y+y*O(y^k)); polcoeff(polcoeff((1-2*X)/(1-X-X*Y*(1-2*X)), n, x), k, y)}

for(n=0, 16, for(k=0, n, print1(T(n, k), ", ")); print(""))

(Maxima) T(n, k):=sum((-2)^i*binomial(k+1, i)*binomial(n-i, k), i, 0, n-k); \\ Vladimir Kruchinin, Nov 02 2011

(Sage)

def A104967_row(n):

@cached_function

def prec(n, k):

if k==n: return 1

if k==0: return 0

return prec(n-1, k-1)-sum(prec(n-i, k-1) for i in (2..n-k+1))

return [prec(n, k) for k in (1..n)]

for n in (1..10): print(A104967_row(n)) # Peter Luschny, Mar 16 2016

(Magma)

A104967:= func< n, k | (&+[(-2)^j*Binomial(k+1, j)*Binomial(n-j, k): j in [0..n-k]]) >;

[A104967(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 09 2021

CROSSREFS

Cf. A078011, A090132, A104968, A104969, A134824, A153881.

Cf. A347171 (rows reversed, up to signs).

Sequence in context: A073266 A125692 A128258 * A098495 A175432 A204118

Adjacent sequences: A104964 A104965 A104966 * A104968 A104969 A104970

KEYWORD

sign,tabl

AUTHOR

Paul D. Hanna, Mar 30 2005

STATUS

approved

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Last modified December 4 07:38 EST 2022. Contains 358544 sequences. (Running on oeis4.)