A052960 -id:A052960 - cp's OEIS frontend

A165241 Triangle T(n,k), 0 <= k <= n, read by rows, given by [1,1,0,0,0,0,0,0,0,...] DELTA [1,0,1,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.

1, 1, 1, 2, 3, 1, 4, 9, 6, 1, 8, 24, 25, 10, 1, 16, 60, 85, 55, 15, 1, 32, 144, 258, 231, 105, 21, 1, 64, 336, 728, 833, 532, 182, 28, 1, 128, 768, 1952, 2720, 2241, 1092, 294, 36, 1, 256, 1728, 5040, 8280, 8361, 5301, 2058, 450, 45, 1

Offset: 0

Philippe Deléham, Sep 09 2009

Rows sums: A006012; Diagonal sums: A052960.

The sums of each column of A117317 with its subsequent column, treated as a lower triangular matrix with an initial null column attached, or, equivalently, the products of the row polynomials p(n,y) of A117317 with (1+y) with the initial first row below added to the final result. The reversal of A117317 is A056242 with several combinatorial interpretations. - Tom Copeland, Jan 08 2017

			Triangle begins:
  1;
  1,  1;
  2,  3,  1;
  4,  9,  6,  1;
  8, 24, 25, 10,  1; ...

Cf. A000012, A000217, A005582, A011782, A084858.

Cf. A056242, A117317.

Sum_{k=0..n} T(n,k)*x^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 = -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, respectively. Sum_{k=0..n} T(n,k)*x^(n-k) = A123335(n), A000007(n), A000012(n), A006012(n), A084120(n), A090965(n), A165225(n), A165229(n), A165230(n), A165231(n), A165232(n) for x = -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, respectively.
G.f.: (1-(1+y)*x)/(1-2(1+y)*x+(y+y^2)*x^2). - Philippe Deléham, Dec 19 2011
T(n,k) = 2*T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k-1) - T(n-2,k-2) with T(0,0) = T(1,0) = T(1,1) = 1 and T(n,k) = 0 if k<0 or if nPhilippe Deléham, Dec 19 2011

O.g.f. corrected by Tom Copeland, Jan 15 2017

A190215 Riordan matrix ((1-x-x^2)/(1-2x-x^2),(x-x^2-x^3)/(1-2x-x^2)).

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 5, 5, 3, 1, 12, 14, 9, 4, 1, 29, 38, 28, 14, 5, 1, 70, 102, 84, 48, 20, 6, 1, 169, 271, 246, 157, 75, 27, 7, 1, 408, 714, 707, 496, 265, 110, 35, 8, 1, 985, 1868, 2001, 1526, 896, 417, 154, 44, 9, 1, 2378, 4858, 5592, 4596, 2930, 1500, 623, 208, 54, 10, 1, 5741, 12569, 15461, 13602, 9330, 5186, 2373, 894, 273, 65, 11, 1

Offset: 0

Emanuele Munarini, May 10 2011

Row sums = A052963.
Diagonal sums = A052960.
Central coefficients = A190315.

			Triangle begins:
    1;
    1,   1;
    2,   2,   1;
    5,   5,   3,   1;
   12,  14,   9,   4,   1;
   29,  38,  28,  14,   5,   1;
   70, 102,  84,  48,  20,   6,   1;
  169, 271, 246, 157,  75,  27,   7,   1;
  408, 714, 707, 496, 265, 110,  35,   8,   1;

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A052963, A052960, A190315.

Flatten[Table[Sum[Binomial[i+k,k]Sum[Binomial[i+j-1,j]Binomial[j,n-k-i-j],{j,0,n-k-i}],{i,0,n-k}],{n,0,12},{k,0,n}]]

create_list(sum(binomial(i+k,k)*sum(binomial(i+j-1,j)*binomial(j,n-k-i-j),j,0,n-k-i),i,0,n-k),n,0,12,k,0,n);

for(n=0,10, for(k=0,n, print1(sum(j=0,n-k, binomial(j+k,k)* sum(r=0,n-k-j, binomial(j+r-1,r)*binomial(r,n-k-j-r))), ", "))) \\ G. C. Greubel, Dec 27 2017

T(n,k) = Sum_{i=0..n-k} (binomial(i+k,k)*Sum_{j=0..n-k-i} (binomial(i+j-1,j)*binomial(j,n-k-i-j) )).

Recurrence: T(n+3,k+1) = 2 T(n+2,k+1) + T(n+2,k) + T(n+1,k+1) - T(n+1,k) - T(n,k).

A112973 Riordan array (1/(1-x-x^2), x(1+x)/(1-x-x^2)^2).

Original entry on oeis.org

1, 1, 1, 2, 4, 1, 3, 12, 7, 1, 5, 31, 31, 10, 1, 8, 73, 110, 59, 13, 1, 13, 162, 340, 267, 96, 16, 1, 21, 344, 956, 1022, 529, 142, 19, 1, 34, 707, 2507, 3479, 2416, 923, 197, 22, 1, 55, 1416, 6231, 10850, 9657, 4900, 1476, 261, 25, 1, 89, 2778, 14840, 31606, 34905

Offset: 0

Paul Barry, Oct 07 2005

Row sums are A091702. Diagonal sums are A052960. First column is A000045(n+1).

Second column is A129707. - Ralf Stephan, Dec 31 2013

			Rows begin
1;
1,1;
2,4,1;
3,12,7,1;
5,31,31,10,1;
8,73,110,59,13,1;

T(n,k):=sum(binomial(m,n-m)*binomial(m+k,2*k),m,floor(n/2),n); /* Vladimir Kruchinin, Apr 21 2015 */

T(n,k) = 2*T(n-1,k) + T(n-1,k-1) + T(n-2,k) + T(n-2,k-1) - 2*T(n-3,k) - T(n-4,k), T(0,0) = T(1,0) = T(1,1) = T(2,2) = 1, T(2,0) = 2, T(2,1) = 4, T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Jan 21 2014
G.f.: (x^2+x-1)/((x^2+x)*y-x^4-2*x^3+x^2+2*x-1). - Vladimir Kruchinin, Apr 21 2015
T(n,k) = Sum_{m=floor(n/2)..n} C(m,n-m)*C(m+k,2*k). - Vladimir Kruchinin, Apr 21 2015

Definition corrected by Ralf Stephan, Dec 31 2013

cp's OEIS Frontend

A165241 Triangle T(n,k), 0 <= k <= n, read by rows, given by [1,1,0,0,0,0,0,0,0,...] DELTA [1,0,1,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.

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A190215 Riordan matrix ((1-x-x^2)/(1-2x-x^2),(x-x^2-x^3)/(1-2x-x^2)).

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A112973 Riordan array (1/(1-x-x^2), x(1+x)/(1-x-x^2)^2).

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