cp's OEIS Frontend

A056241 Triangle T(n,k) = number of k-part order-consecutive partitions of n (1<=k<=n).

%I A056241 #48 Jul 02 2025 16:02:00
%S A056241 1,1,1,1,3,1,1,6,6,1,1,10,19,10,1,1,15,45,45,15,1,1,21,90,141,90,21,1,
%T A056241 1,28,161,357,357,161,28,1,1,36,266,784,1107,784,266,36,1,1,45,414,
%U A056241 1554,2907,2907,1554,414,45,1,1,55,615,2850,6765,8953,6765,2850,615,55
%N A056241 Triangle T(n,k) = number of k-part order-consecutive partitions of n (1<=k<=n).
%C A056241 Forms the even-indexed trinomial coefficients (A027907). Matrix inverse is A104027. - _Paul D. Hanna_, Feb 26 2005
%C A056241 Subtriangle (for 1<=k<=n) of triangle defined by [0, 1, 0, 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_, Oct 29 2006
%H A056241 Jean-Luc Baril, Pamela E. Harris, and José L. Ramírez, <a href="https://arxiv.org/abs/2405.05357">Flattened Catalan Words</a>, arXiv:2405.05357 [math.CO], 2024. See p. 6.
%H A056241 Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Barry/barry321.html">Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices</a>, Journal of Integer Sequences, 19, 2016, #16.3.5.
%H A056241 Karl Dilcher and Larry Ericksen, <a href="https://arxiv.org/abs/2405.12024">Polynomials and algebraic curves related to certain binary and b-ary overpartitions</a>, arXiv:2405.12024 [math.CO], 2024. See p. 7.
%H A056241 F. K. Hwang and C. L. Mallows, <a href="http://dx.doi.org/10.1016/0097-3165(95)90097-7">Enumerating nested and consecutive partitions</a>, J. Combin. Theory Ser. A 70 (1995), no. 2, 323-333.
%F A056241 T(n, k) = Sum_{j=0..k-1} C(n-1, 2k-j-2)*C(2k-j-2, j).
%F A056241 G.f.: A(x, y) = (1 - x*(1+y))/(1 - 2*x*(1+y) + x^2*(1+y+y^2)) (offset=0). - _Paul D. Hanna_, Feb 26 2005
%F A056241 Sum_{k, 1<=k<=n}T(n,k)=A124302(n). Sum_{k, 1<=k<=n}(-1)^(n-k)*T(n,k)=A117569(n). - _Philippe Deléham_, Oct 29 2006
%F A056241 From _Paul Barry_, Sep 28 2010: (Start)
%F A056241 G.f.: 1/(1-x-xy-x^2y/(1-x-xy)).
%F A056241 E.g.f.: exp((1+y)x)*cosh(sqrt(y)*x).
%F A056241 T(n,k) = Sum_{j=0..n} C(n,j)*C(n-j,2*(k-j)). (End)
%F A056241 T(n,k) = 2*T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k) - T(n-2,k-1) - T(n-2,k-2), T(1,1) = T(2,1) = T(2,2) = 1, T(n,k) = 0 if k<1 or if k>n. - _Philippe Deléham_, Mar 27 2014
%e A056241 Triangle begins:
%e A056241   1;
%e A056241   1,1;
%e A056241   1,3,1;
%e A056241   1,6,6,1;
%e A056241   1,10,19,10,1;
%e A056241   ...
%e A056241 Triangle (0, 1, 0, 1, 0, 0, 0...) DELTA (1, 0, 1, 0, 0, 0, ...) begins:
%e A056241   1;
%e A056241   0, 1;
%e A056241   0, 1, 1;
%e A056241   0, 1, 3, 1;
%e A056241   0, 1, 6, 6, 1;
%e A056241   0, 1, 10, 19, 10, 1;
%e A056241   0, 1, 15, 45, 45, 15, 1;
%e A056241   0, 1, 21, 90, 141, 90, 21, 1;
%e A056241   ... - _Philippe Deléham_, Mar 27 2014
%t A056241 t[n_, k_] := Sum[ Binomial[n, j]*Binomial[n-j, 2*(k-j)], {j, 0, n}]; Flatten[ Table[t[n, k], {n, 0, 10}, {k, 0, n}]] (* _Jean-François Alcover_, Oct 11 2011, after _Paul Barry_ *)
%o A056241 (PARI) T(n,k)=if(n<k || k<1,0,polcoeff((1+x+x^2)^(n-1)+O(x^(2*k)),2*k-2)) \\ _Paul D. Hanna_
%Y A056241 Columns are A000217, A005712, A005714, A005716.
%Y A056241 Cf. A027907, A104027, A117569, A124302.
%K A056241 nonn,tabl,easy,nice
%O A056241 1,5
%A A056241 _Colin Mallows_, Aug 23 2000
%E A056241 More terms from _James Sellers_, Aug 25 2000
%E A056241 More terms from _Paul D. Hanna_, Feb 26 2005