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A281000
Triangle read by rows: T(n,k) = binomial(2*n+1, 2*k+1)*binomial(2*n-2*k, n-k)/(n+1-k) for 0 <= k <= n.
1
1, 3, 1, 10, 10, 1, 35, 70, 21, 1, 126, 420, 252, 36, 1, 462, 2310, 2310, 660, 55, 1, 1716, 12012, 18018, 8580, 1430, 78, 1, 6435, 60060, 126126, 90090, 25025, 2730, 105, 1, 24310, 291720, 816816, 816816, 340340, 61880, 4760, 136, 1, 92378, 1385670, 4988412, 6651216, 3879876, 1058148, 135660, 7752, 171, 1
OFFSET
0,2
LINKS
FORMULA
T(n,k) = A097610(2*n+1, 2*k+1) = binomial(2*n+1, 2*k+1)*A000108(n-k) = A280580(n,k)*(2*n+1)/(2*k+1) for 0 <= k <= n.
Recurrences: T(n,0) = (2*n+1)*A000108(n) and
(1) T(n,k) = T(n,k-1)*(n+1-k)*(n+2-k)/(2*k*(2*k+1)) for 0 < k <= n,
(2) T(n,k) = T(n-1, k-1)*n*(2*n+1)/(k*(2*k+1)).
The row polynomials p(n,x) = Sum_{k=0..n} T(n,k)*x^(2*k+1) satisfy the recurrence equation p"(n,x) = (2*n+1)*2*n*p(n-1,x) with initial value p(0,x) = x (n > 0, p" is the second derivative of p), and Sum_{n>=0} p(n,x)*t^(2*n+1)/ ((2*n+1)!) = sinh(x*t)*(Sum_{n>=0} A000108(n)*t^(2*n)/((2*n)!)).
Conjectures:
(1) Antidiagonal sums equal A001003(n+1);
(2) Sum_{k=0..n} (-1)^k*T(n,k)*(2*k+1)*A000108(k)*A103365(k) = A000007(n);
(3) Matrix inverse equals T(n,k)*A103365(n+1-k).
Sum_{k=0..n} (n+1-k)*T(n,k) = A002426(2*n+1) = A273055(n).
Sum_{k=0..n} T(n,k)*(2*k+1)*A000108(k) = (2*n+1)*A000108(n)*A000108(n+1) = A125558(n+1).
Matrix product: Sum_{i=0..n} T(n,i)*T(i,k) = T(n,k)*A000108(n+1-k) for 0<=k<=n.
EXAMPLE
Triangle begins:
n\k: 0 1 2 3 4 5 6 7 8 . . .
0: 1
1: 3 1
2: 10 10 1
3: 35 70 21 1
4: 126 420 252 36 1
5: 462 2310 2310 660 55 1
6: 1716 12012 18018 8580 1430 78 1
7: 6435 60060 126126 90090 25025 2730 105 1
8: 24310 291720 816816 816816 340340 61880 4760 136 1
etc.
T(3,2) = binomial(7,5) * binomial(2,1) / (3+1-2) = 21 * 2 / 2 = 21. - Indranil Ghosh, Feb 15 2017
MATHEMATICA
Table[Binomial[2n+1, 2k+1] Binomial[2n-2k, n-k]/(n+1-k), {n, 0, 10}, {k, 0, n}]// Flatten (* Harvey P. Dale, Nov 25 2018 *)
CROSSREFS
Row sums are A099250.
Triangle related to A000108, A097610, A280580.
Sequence in context: A144697 A185419 A252501 * A146154 A068438 A064060
KEYWORD
nonn,easy,tabl
AUTHOR
Werner Schulte, Jan 12 2017
STATUS
approved