A001875 -id:A001875 - cp's OEIS frontend

A152440 Riordan matrix (1/(1-x-x^2),x/(1-x-x^2)^2).

1, 1, 1, 2, 3, 1, 3, 9, 5, 1, 5, 22, 20, 7, 1, 8, 51, 65, 35, 9, 1, 13, 111, 190, 140, 54, 11, 1, 21, 233, 511, 490, 255, 77, 13, 1, 34, 474, 1295, 1554, 1035, 418, 104, 15, 1, 55, 942, 3130, 4578, 3762, 1925, 637, 135, 17, 1, 89, 1836, 7285, 12720, 12573, 7865, 3276

Offset: 0

Emanuele Munarini, Dec 04 2008, Dec 05 2008

From Philippe Deléham, Feb 20 2014: (Start)
T(n,0)   = A000045(n+1);
T(n+1,1) = A001628(n);
T(n+2,2) = A001873(n);
T(n+3,3) = A001875(n).
Row sums are A238236(n). (End)

			Triangle begins:
1;
1, 1;
2, 3, 1;
3, 9, 5, 1;
5, 22, 20, 7, 1;
8, 51, 65, 35, 9, 1;
13, 111, 190, 140, 54, 11, 1;
21, 233, 511, 490, 255, 77, 13, 1, etc.
- _Philippe Deléham_, Feb 20 2014

The first row is given by A000045.

Cf. A037027, A238241.

a(n,k) = sum( binomial(n-j-k,2k) binomial(n-j-k,j), j=0...(n-k)/2 )
a(n,k) = sum( binomial(i+2k,2k) binomial(n-i+k,i+2k), i=0...(n - k)/2 )
Recurrence: a(n+4,k+1) - 2 a(n+3,k+1) - a(n+3,k) - a(n+2,k+1) + 2 a(n+1,k+1) + a(n,k+1) = 0
GF for columns: 1/(1-x-x^2)(x/(1-x-x^2)^2)^k
GF: (1-x-x^2)/((1-x-x^2)^2-xy)
T(n,k) = A037027(n+k, 2*k). - Philippe Deléham, Feb 20 2014

A132883 Triangle read by rows: T(n,k) is the number of paths in the first quadrant from (0,0) to (n,0), consisting of steps U=(1,1), D=(1,-1), h=(1,0) and H=(2,0), having k U steps (0 <= k <= floor(n/2)).

Original entry on oeis.org

1, 1, 2, 1, 3, 3, 5, 9, 2, 8, 22, 10, 13, 51, 40, 5, 21, 111, 130, 35, 34, 233, 380, 175, 14, 55, 474, 1022, 700, 126, 89, 942, 2590, 2450, 756, 42, 144, 1836, 6260, 7770, 3570, 462, 233, 3522, 14570, 22890, 14490, 3234, 132, 377, 6666, 32870, 63600, 52668

Offset: 0

Emeric Deutsch, Sep 03 2007

Row n has 1+floor(n/2) terms. T(n,0) = A000045(n+1) (the Fibonacci numbers). T(2n,n) = binomial(2n,n)/(n+1) = A000108(n) (the Catalan numbers). Row sums yield A118720. Column k has g.f. = c(k)z^(2k)/(1-z-z^2)^(2k+1), where c(k) = binomial(2k,k)/(k+1) are the Catalan numbers; accordingly, T(n,1) = A001628(n-2), T(n,2) = 2*A001873(n-4), T(n,3) = 5*A001875(n-6). Sum_{k>=0} k*T(n,k) = A106050(n+1).

			Triangle starts:
   1;
   1;
   2,  1;
   3,  3;
   5,  9,  2;
   8, 22, 10;
  13, 51, 40,  5;
T(3,1)=3 because we have hUD, UhD and UDh.

Cf. A000045, A118720, A106050, A000108.

G:=((1-z-z^2-sqrt(1-2*z-z^2+2*z^3+z^4-4*t*z^2))*1/2)/(t*z^2): Gser:=simplify(series(G, z = 0, 17)): for n from 0 to 13 do P[n]:=sort(coeff(Gser,z,n)) end do: for n from 0 to 13 do seq(coeff(P[n],t,j),j=0..floor((1/2)*n)) end do; # yields sequence in triangular form

G.f.: G = G(t,z) satisfies G = 1 + zG + z^2*G + tz^2*G^2 (see explicit expression at the Maple program).

A182880 Triangle read by rows: T(n,k) is the number of weighted lattice paths in L_n having k (1,1)-steps. L_n is the set of lattice paths of weight n that start at (0,0) and end on the horizontal axis and whose steps are of the following four kinds: a (1,0)-step with weight 1; a (1,0)-step with weight 2; a (1,1)-step with weight 2; a (1,-1)-step with weight 1. The weight of a path is the sum of the weights of its steps.

Original entry on oeis.org

1, 1, 2, 3, 2, 5, 6, 8, 18, 13, 44, 6, 21, 102, 30, 34, 222, 120, 55, 466, 390, 20, 89, 948, 1140, 140, 144, 1884, 3066, 700, 233, 3672, 7770, 2800, 70, 377, 7044, 18780, 9800, 630, 610, 13332, 43710, 31080, 3780, 987, 24946, 98610, 91560, 17850, 252, 1597, 46218, 216732, 254400, 72450, 2772

Offset: 0

Emeric Deutsch, Dec 11 2010

Sum of entries in row n is A051286(n).

			T(3,1)=2. Indeed, denoting by h (H) the (1,0)-step of weight 1 (2), and u=(1,1), d=(1,-1), the five paths of weight 3 are ud, du, hH, Hh, and hhh; two of them, ud and du, have exactly one u step.
Triangle starts:
   1;
   1;
   2;
   3,  2;
   5,  6;
   8, 18;
  13, 44, 6;

M. Bona and A. Knopfmacher, On the probability that certain compositions have the same number of parts, Ann. Comb., 14 (2010), 291-306.
E. Munarini, N. Zagaglia Salvi, On the rank polynomial of the lattice of order ideals of fences and crowns, Discrete Mathematics 259 (2002), 163-177.

Cf. A051286, A000045, A182881.

G:=1/sqrt(1-2*z-z^2+2*z^3+z^4-4*t*z^3): Gser:=simplify(series(G,z=0,18)): for n from 0 to 16 do P[n]:=sort(coeff(Gser,z,n)) od: for n from 0 to 16 do seq(coeff(P[n],t,k),k=0..floor(n/3)) od; # yields sequence in triangular form

T(n,0) = A000045(n+1) (the Fibonacci numbers).
Sum_{k=0..n} k*T(n,k) = A182881(n).
G.f.: G(t,z) = 1/sqrt(1 - 2*z - z^2 + 2*z^3 + z^4 - 4*t*z^3).
The g.f. of column k is binomial(2n,n)*z^(3n)/(1-z-z^2)^(2n+1).
Apparently, T(n,1) = 2*A001628(n-3), T(n,2) = 6*A001873(n-6), T(n,3) = 20*A001875(n-9). - R. J. Mathar, Dec 11 2010

A132886 Triangle read by rows: T(n,k) is the number of paths in the right half-plane, from (0,0) to (n,0), consisting of steps U=(1,1), D=(1,-1), h=(1,0) and H=(2,0), having k U steps (0 <= k <= floor(n/2)).

Original entry on oeis.org

1, 1, 2, 2, 3, 6, 5, 18, 6, 8, 44, 30, 13, 102, 120, 20, 21, 222, 390, 140, 34, 466, 1140, 700, 70, 55, 948, 3066, 2800, 630, 89, 1884, 7770, 9800, 3780, 252, 144, 3672, 18780, 31080, 17850, 2772, 233, 7044, 43710, 91560, 72450, 19404, 924, 377, 13332, 98610

Offset: 0

Emeric Deutsch, Sep 03 2007

Row n has 1+floor(n/2) terms. T(n,0) = A000045(n+1) (the Fibonacci numbers). T(2n,n) = binomial(2n,n) = A000984(n) (the central binomial coefficients). Row sums yield A059345. Column k has g.f. = binomial(2k,k)*z^(2k)/(1-z-z^2)^(2k+1); accordingly, T(n,1) = 2*A001628(n-2), T(n,2) = 6*A001873(n-4), T(n,3) = 20*A001875(n-6). See A132883 for the same statistic on paths restricted to the first quadrant.

			Triangle starts:
   1;
   1;
   2,   2;
   3,   6;
   5,  18,   6;
   8,  44,  30;
  13, 102, 120,  20;
T(3,1)=6 because we have hUD, UhD, UDh, hDU, DhU and DUh.

Cf. A000045, A059345, A001628, A001873, A001875, A132883.

G:=1/sqrt((1-z-z^2)^2-4*t*z^2): Gser:=simplify(series(G,z=0,17)): for n from 0 to 13 do P[n]:= sort(coeff(Gser,z,n)) end do: for n from 0 to 13 do seq(coeff(P[n],t,j),j=0..floor((1/2)*n)) end do; # yields sequence in triangular form

G.f.: G(t,z) = 1/sqrt((1-z-z^2)^2 - 4tz^2).

cp's OEIS Frontend

A152440 Riordan matrix (1/(1-x-x^2),x/(1-x-x^2)^2).

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A132883 Triangle read by rows: T(n,k) is the number of paths in the first quadrant from (0,0) to (n,0), consisting of steps U=(1,1), D=(1,-1), h=(1,0) and H=(2,0), having k U steps (0 <= k <= floor(n/2)).

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Maple

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A132886 Triangle read by rows: T(n,k) is the number of paths in the right half-plane, from (0,0) to (n,0), consisting of steps U=(1,1), D=(1,-1), h=(1,0) and H=(2,0), having k U steps (0 <= k <= floor(n/2)).

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Comments

Examples

Crossrefs

Programs

Maple

Formula