0,2

Apart from signs, identical to A053122.

Coefficient array for Morgan-Voyce polynomial B(n,x); see A085478 for references. DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Feb 16 2004

T(n,k)=number of compositions of n having k parts when there are q kinds of part q (q=1,2,...). Example: T(4,2)=10 because we have (1,3),(1,3'),(1,3"), (3,1),(3',1),(3",1),(2,2),(2,2'),(2',2) and (2',2'). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 09 2005

T. D. Noe, Rows n=0..50 of triangle, flattened

G.f.: x*y/(1-(2+y)*x+x^2). To get row n, expand this in powers of x then expand the coefficient of x^n in increasing powers of y.

If indexing begins at 0 we have: T(n, k) = (n+k+1)!/((n-k)!*(2k+1))!. T(n, k) = Sum_{j>=0} T(n-1-j, k-1)*(j+1) with T(n, 0) = n+1, T(n, k) = 0 if n<k. T(n, k) = T(n-1, k-1) + T(n-1, k) + Sum_{j>=0} (-1)^j*T(n-1, k+j)*A000108(j) with T(n, k) = 0 if k<0, T(0, 0)=1 and T(0, k) = 0 for k>0. G.f. for the column k : Sum_{n>=0} T(n, k)*x^n = (x^k)/(1-x)^(2k+2). Row sums : Sum_{k>=0} T(n, k) = A001906(n+1). - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Feb 16 2004

Diagonal sums are A000079(n)=sum{k=0..floor(n/2), binomial(n+k+1, n-k)}. - Paul Barry (pbarry(AT)wit.ie), Jun 21 2004

Riordan array (1/(1-x)^2, x/(1-x)^2). - Paul Barry (pbarry(AT)wit.ie), Oct 22 2006

Triangle begins:

.........................1

........................2,.1

......................3,.4,.1

....................4,.10,.6,.1

..................5,.20,.21,.8,.1

................6,.35,.56,.36,.10,.1

.............7,.56,.126,.120,.55,.12,.1

..........8,.84,.252,.330,.220,.78,.14,.1

for n from 1 to 11 do seq(binomial(n+k-1, 2*k-1), k=1..n) od; # yields sequence in triangular form (Deutsch)

(PARI) T(n, k)=if(n<0, 0, binomial(n+k-1, 2*k-1))

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

This triangle is formed from odd-numbered rows of triangle A011973 read in reverse order.

The column sequences are A000027, A000292, A000389, A000580, A000582, A001288 for k=1..6, resp. For k=7..24 they are A010966..(+2)..A011000 and for k=25..50 they are A017713..(+2)..A017763.

Cf. A053123, A049310. Row sums give A001906.

With signs: A053122.

Sequence in context: A051128 A137614 A053122 this_sequence A104711 A133112 A137649

Adjacent sequences: A078809 A078810 A078811 this_sequence A078813 A078814 A078815

easy,nice,nonn,tabl

Michael Somos, Dec 05, 2002

Edited by njas, Apr 28 2008