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A098158 Triangle T(n,k) with diagonals T(n,n-k) = binomial(n, 2*k). 77
1, 0, 1, 0, 1, 1, 0, 0, 3, 1, 0, 0, 1, 6, 1, 0, 0, 0, 5, 10, 1, 0, 0, 0, 1, 15, 15, 1, 0, 0, 0, 0, 7, 35, 21, 1, 0, 0, 0, 0, 1, 28, 70, 28, 1, 0, 0, 0, 0, 0, 9, 84, 126, 36, 1, 0, 0, 0, 0, 0, 1, 45, 210, 210, 45, 1, 0, 0, 0, 0, 0, 0, 11, 165, 462, 330, 55, 1, 0, 0, 0, 0, 0, 0, 1, 66, 495, 924 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,9

COMMENTS

Row sums are A011782. Inverse is A065547.

Triangle T(n,k), 0<=k<=n, read by rows, given by [0, 1, -1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 1, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938 . - Philippe Deléham, Jul 29 2006

Sum of entries in column k is A001519(k+1)(the odd indexed Fibonacci numbers). - Philippe Deléham, Dec 02 2008

LINKS

G. C. Greubel, Rows n = 0..49 of triangle, flattened

D. Dumont and J. Zeng, Polynomes d'Euler et les fractions continues de Stieltjes-Rogers, Ramanujan J. 2 (1998) 3, 387-410.

FORMULA

Triangle T(n, k) = binomial(n, 2(n-k)).

From Tom Copeland, Oct 10 2016: (Start)

E.g.f.: exp(t*x) * cosh(t*sqrt(x)).

O.g.f.: (1/2) * [ 1 / [1 - (1 + sqrt(1/x))*x*t] + 1 / [1 - (1 - sqrt(1/x))*x*t] ].

Row polynomial: x^n * [(1 + sqrt(1/x))^n + (1 - sqrt(1/x))^n] / 2. (End)

Column k is generated by the polynomial Sum_{j=0..floor(k/2)} C(k, 2j) * x^(k-j). - Paul Barry, Jan 22 2005

G.f.: (1-x*y)/((1-x*y)^2 - x^2*y). - Paul D. Hanna, Feb 25 2005

Sum_{k, 0<=k<=n}x^k*T(n,k)= A009116(n), A000007(n), A011782(n), A006012(n), A083881(n), A081335(n), A090139(n), A145301(n), A145302(n), A145303(n), A143079(n) for x = -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 respectively . - Philippe Deléham, Dec 04 2006, Oct 15 2008, Oct 19 2008

T(n,k) = T(n-1,k-1) + Sum_{i=0..k-1} T(n-2-i,k-1-i); T(0,0)=1; T(n,k)=0 if n<0, if k<0, if n<k . E.g. : T(8,5) =T(7,4) +T(6,4) +T(5,3) +T(4,2) +T(3,1) +T(2,0) = 7+15+5+1+0+0 = 28. - Philippe Deléham, Dec 04 2006

Sum_{k=0..n} T(n,k)*x^(n-k) = A000012(n), A011782(n), A001333(n), A026150(n), A046717(n), A084057(n), A002533(n), A083098(n), A084058(n), A003665(n), A002535(n), A133294(n), A090042(n), A125816(n), A133343(n), A133345(n), A120612(n), A133356(n), A125818(n) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18 respectively . - Philippe Deléham, Dec 24 2007

Sum_{k, 0<=k<=n}T(n,k)*(-x)^(n-k) = A000012(n), A146559(n), A087455(n), A138230(n), A006495(n), A138229(n) for x = 0,1,2,3,4,5 respectively. - Philippe Deléham, Nov 14 2008

T(n,k) = A085478(k,n-k). - Philippe Deléham, Dec 02 2008

T(n,k) = 2*T(n-1,k-1) + T(n-2,k-1) - T(n-2,k-2), T(0,0) = T(1,1) = 1, T(1,0) = 0 and T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Mar 15 2012

EXAMPLE

Rows begin

  1;

  0, 1;

  0, 1, 1;

  0, 0, 3, 1;

  0, 0, 1, 6, 1;

MATHEMATICA

Table[Binomial[n, 2*(n-k)], {n, 0, 12}, {k, 0, n}]//Flatten (* Michael De Vlieger, Oct 12 2016 *)

PROG

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

(PARI) T(n, k) = binomial(n, 2*(n-k));

for(n=0, 12, for(k=0, n, print1(T(n, k), ", "))) \\ G. C. Greubel, Aug 01 2019

(MAGMA) [Binomial(n, 2*(n-k)): k in [0..n], n in [0..12]]; // G. C. Greubel, Aug 01 2019

(Sage) [[binomial(n, 2*(n-k)) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Aug 01 2019

(GAP) Flat(List([0..12], n-> List([0..n], k-> Binomial(n, 2*(n-k)) ))); # G. C. Greubel, Aug 01 2019

CROSSREFS

Cf. A098157, A034839.

Cf. A119900. - Philippe Deléham, Dec 02 2008

Sequence in context: A247505 A117389 A122083 * A110319 A036872 A036871

Adjacent sequences:  A098155 A098156 A098157 * A098159 A098160 A098161

KEYWORD

easy,nonn,tabl

AUTHOR

Paul Barry, Aug 29 2004

STATUS

approved

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Last modified October 23 20:17 EDT 2019. Contains 328373 sequences. (Running on oeis4.)