A003480 -id:A003480 - cp's OEIS frontend

A181336 Triangle read by rows: T(n,k) is the number of 2-compositions of n having k even entries in the top row. A 2-composition of n is a nonnegative matrix with two rows, such that each column has at least one nonzero entry and whose entries sum up to n.

1, 1, 1, 2, 4, 1, 5, 11, 7, 1, 11, 31, 29, 10, 1, 25, 83, 102, 56, 13, 1, 56, 217, 329, 245, 92, 16, 1, 126, 556, 1000, 938, 487, 137, 19, 1, 283, 1403, 2917, 3292, 2180, 855, 191, 22, 1, 636, 3498, 8247, 10865, 8740, 4406, 1376, 254, 25, 1, 1429, 8636, 22756, 34248

Offset: 0

Emeric Deutsch, Oct 14 2010

The sum of entries in row n is A003480(n).
T(n,0)=A006054(n+1) (n>=1).
Sum(k*T(n,k), k>=0)=A181337(n).
For the statistic "number of odd entries in the top row" see A181304.

			T(2,1)=4 because we have (0/2), (2/0), (1,0/0,1), and (0,1/1,0) (the 2-compositions are written as (top row / bottom row)).
Triangle starts:
1;
1,1;
2,4,1;
5,11,7,1;
11,31,29,10,1;
25,83,102,56,13,1;

G. Castiglione, A. Frosini, E. Munarini, A. Restivo and S. Rinaldi, Combinatorial aspects of L-convex polyominoes, European Journal of Combinatorics, 28, 2007, 1724-1741.

Cf. A003480, A006054, A181304, A181337,

G := (1+z)*(1-z)^2/(1-2*z-z^2+z^3-s*z*(1+z-z^2)): Gser := simplify(series(G, z = 0, 13)): for n from 0 to 10 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 10 do seq(coeff(P[n], s, k), k = 0 .. n) end do; # yields sequence in triangular form

G.f.=G(s,z)=(1+z)(1-z)^2/[1-2z-z^2+z^3-sz(1+z-z^2)].
The g.f. of column k is z^k*(1+z)(1-z)^2*(1+z-z^2)^k/(1-2z-z^2+z^3)^{k+1} (we have a Riordan array).
The g.f. H=H(t,s,z), where z marks size and t (s) marks odd (even) entries in the top row, is given by H = (1+z)(1-z)^2/[(1+z)(1-z)^2-(t+s)z-sz^2*(1-z)].

A259860 a(n+8)+34*a(n+4)+a(n)=0 with a(0)-a(7) as shown.

Original entry on oeis.org

1, 5, 12, 12, -29, -169, -408, -408, 985, 5741, 13860, 13860, -33461, -195025, -470832, -470832, 1136689, 6625109, 15994428, 15994428, -38613965, -225058681, -543339720, -543339720, 1311738121, 7645370045, 18457556052, 18457556052, -44560482149

Offset: 0

N. J. A. Sloane, Jul 08 2015

Colin Barker, Table of n, a(n) for n = 0..1000
J. Riordan, The distribution of crossings of chords joining pairs of 2n points on a circle, Math. Comp., 29 (1975), 215-222.
J. Riordan, The distribution of crossings of chords joining pairs of 2n points on a circle, Math. Comp., 29 (1975), 215-222. [Annotated scanned copy]
Index entries for linear recurrences with constant coefficients, signature (4,-8,4,-1).

Cf. A259861.

A259860 := proc(n)
    if n <= 7 then
        op(n+1,[1, 5, 12, 12, -29, -169, -408, -408, 985, 5741, 13860, 13860]) ;
    else
        -34*procname(n-4)-procname(n-8) ;
    fi ;
end proc: # R. J. Mathar, Jun 24 2024

LinearRecurrence[{4,-8,4,-1},{1,5,12,12},40] (* Harvey P. Dale, Mar 14 2016 *)

Vec((x+1)/(x^4-4*x^3+8*x^2-4*x+1) + O(x^50)) \\ Colin Barker, Jul 09 2015

G.f.: (x+1) / (x^4-4*x^3+8*x^2-4*x+1). - Colin Barker, Jul 09 2015

A259861 a(n+8)+34*a(n+4)+a(n)=0 with a(0)-a(7) as shown.

Original entry on oeis.org

2, 2, -5, -29, -70, -70, 169, 985, 2378, 2378, -5741, -33461, -80782, -80782, 195025, 1136689, 2744210, 2744210, -6625109, -38613965, -93222358, -93222358, 225058681, 1311738121, 3166815962, 3166815962, -7645370045, -44560482149, -107578520350, -107578520350

Offset: 0

N. J. A. Sloane, Jul 08 2015

Colin Barker, Table of n, a(n) for n = 0..1000
J. Riordan, The distribution of crossings of chords joining pairs of 2n points on a circle, Math. Comp., 29 (1975), 215-222.
J. Riordan, The distribution of crossings of chords joining pairs of 2n points on a circle, Math. Comp., 29 (1975), 215-222. [Annotated scanned copy]
Index entries for linear recurrences with constant coefficients, signature (4,-8,4,-1).

Cf. A259860.

Vec(-(x^3-3*x^2+6*x-2)/(x^4-4*x^3+8*x^2-4*x+1) + O(x^50)) \\ Colin Barker, Jul 09 2015

G.f.: -(x^3-3*x^2+6*x-2) / (x^4-4*x^3+8*x^2-4*x+1). - Colin Barker, Jul 09 2015

A128611 Number of Z-convex polyominoes with semiperimeter n.

Original entry on oeis.org

0, 0, 1, 2, 7, 28, 116, 484, 2022, 8448, 35290, 147376, 615228, 2567060, 10704976, 44611804, 185780308, 773060804, 3214225836, 13352979316, 55426067494, 229870371888, 952548347122, 3943943111920, 16316243701350, 67447113649312, 278592165886198, 1149863118820584, 4742473257979906, 19545876370622104, 80502059920697442

Offset: 0

Ralf Stephan, May 08 2007

nonn

Robert Israel, Table of n, a(n) for n = 0..1658
Adrien Boussicault, Simone Rinaldi, and Samanta Socci, The number of directed k-convex polyominoes, arXiv preprint arXiv:1501.00872 [math.CO], 2015; Discrete Math., 343 (2020), #111731, 22 pages. See page 2.
E. Duchi, S. Rinaldi and G. Schaeffer, The number of Z-convex polyominoes, arXiv:math/0602124 [math.CO], 2006.

Cf. A003480, A005436.

d:=(1-2*t-sqrt(1-4*t))/2:
t1:=
2*t^4*(1-2*t)^2*d/( (1-4*t)^2*(1-3*t)*(1-t) )
+ t^2*(1-6*t+10*t^2-2*t^3-t^4)/( (1-4*t)*(1-3*t)*(1-t) ):
series(t1,t,120):
seriestolist(%); # N. J. A. Sloane, Oct 02 2011

gf = 2 t^4 (1-2t)^2 d/((1-4t)^2 (1-3t)(1-t)) + t^2 (1-6t+10t^2-2t^3-t^4)/ ((1-4t)(1-3t)(1-t)) /. d -> (1-2t-Sqrt[1-4t])/2;
CoefficientList[gf + O[t]^31, t] (* Jean-François Alcover, Aug 17 2018 *)

The Duchi paper has a g.f.
Asymptotically, a(n) ~ n/24 * 4^n.
G.f.: Let d:=(1-2*t-sqrt(1-4*t))/2; then g.f. is 2*t^4*(1-2*t)^2*d/( (1-4*t)^2*(1-3*t)*(1-t) ) + t^2*(1-6*t+10*t^2-2*t^3-t^4)/( (1-4*t)*(1-3*t)*(1-t) ). - N. J. A. Sloane, Oct 02 2011
(-960+384*n)*a(n)+(1760-992*n)*a(n+1)+(-924+984*n)*a(2+n)+(64-490*n)*a(n+3)+(82+131*n)*a(n+4)+(-24-18*n)*a(n+5)+(2+n)*a(n+6), a(0) = 0. - Robert Israel, Aug 17 2018

A181293 Triangle read by rows: T(n,k) is the number of 2-compositions of n having k 0's (0<=k<=n) A 2-composition of n is a nonnegative matrix with two rows, such that each column has at least one nonzero entry and whose entries sum up to n.

Original entry on oeis.org

1, 0, 2, 1, 2, 4, 2, 6, 8, 8, 4, 14, 24, 24, 16, 8, 32, 64, 80, 64, 32, 16, 72, 164, 240, 240, 160, 64, 32, 160, 408, 680, 800, 672, 384, 128, 64, 352, 992, 1848, 2480, 2464, 1792, 896, 256, 128, 768, 2368, 4864, 7296, 8288, 7168, 4608, 2048, 512, 256, 1664, 5568

Offset: 0

Emeric Deutsch, Oct 12 2010

The sum of entries in row n is A003480(n).

Sum_{k=0..n} k*T(n,k) = A181294(n).

			T(2,0)=1, T(2,1)=2, T(2,2)=4 because the 2-compositions of 2, written as (top row/bottom row), are (1/1), (0/2), (2/0), (1,0/0,1), (0,1/1,0), (1,1/0,0), (0,0/1,1).
Triangle starts:
1;
0,2;
1,2,4;
2,6,8,8;
4,14,24,24,16;
...

G. Castiglione, A. Frosini, E. Munarini, A. Restivo and S. Rinaldi, Combinatorial aspects of L-convex polyominoes, European J. Combin. 28 (2007), no. 6, 1724-1741.

Cf. A003480, A181294.

G := (1-z)^2/(1-2*z-2*t*z+2*t*z^2): Gser := simplify(series(G, z = 0, 14)): for n from 0 to 10 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 10 do seq(coeff(P[n], t, k), k = 0 .. n) end do; # yields sequence in triangular form

G.f.: G(t,z) = (1-z)^2/(1-2*z-2*t*z+2*t*z^2).

G.f. of column k is 2^k*z^k*(1-z)^{k+2}/(1-2*z)^{k+1} (we have a Riordan array).

A181302 Triangle read by rows: T(n,k) is the number of 2-compositions of n having k columns with distinct entries (0<=k<=n).

Original entry on oeis.org

1, 0, 2, 1, 2, 4, 0, 8, 8, 8, 2, 8, 32, 24, 16, 0, 24, 56, 104, 64, 32, 4, 24, 152, 248, 304, 160, 64, 0, 64, 248, 712, 896, 832, 384, 128, 8, 64, 568, 1496, 2800, 2880, 2176, 896, 256, 0, 160, 888, 3560, 6976, 9824, 8576, 5504, 2048, 512, 16, 160, 1848, 6904, 17904

Offset: 0

Emeric Deutsch, Oct 13 2010

A 2-composition of n is a nonnegative matrix with two rows, such that each column has at least one nonzero entry and whose entries sum up to n.

			T(2,1) = 2 because we have (0/2) and (2/0) (the 2-compositions are written as (top row/bottom row)).
Triangle starts:
  1;
  0,2;
  1,2,4;
  0,8,8,8;
  2,8,32,24,16;

G. Castiglione, A. Frosini, E. Munarini, A. Restivo and S. Rinaldi, Combinatorial aspects of L-convex polyominoes, European J. Combin. 28 (2007), no. 6, 1724-1741.

Cf. A003480, A181296.

G := (1+z)*(1-z)^2/((1-z)*(1-2*z^2)-2*t*z): Gser := simplify(series(G, z = 0, 15)): for n from 0 to 10 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 10 do seq(coeff(P[n], t, k), k = 0 .. n) end do; # yields sequence in triangular form

G.f.: G(t,z) = (1+z)*(1-z)^2/((1-z)*(1-2*z^2)-2*t*z).
G.f. of column k: 2^k*z^k*(1+z)/((1-2*z^2)^(k+1)*(1-z)^(k-1)) (we have a Riordan array).
Sum_{k>=0} k*T(n,k) = A181296(n).
Sum_{k>=0} T(n,k) = A003480(n).

A181338 Triangle read by rows: T(n,k) is the number of 2-compositions of n having largest entry k (1<=k<=n).

Original entry on oeis.org

2, 5, 2, 12, 10, 2, 29, 41, 10, 2, 70, 152, 46, 10, 2, 169, 536, 193, 46, 10, 2, 408, 1830, 770, 198, 46, 10, 2, 985, 6120, 2972, 811, 198, 46, 10, 2, 2378, 20178, 11202, 3218, 816, 198, 46, 10, 2, 5741, 65867, 41481, 12484, 3259, 816, 198, 46, 10, 2, 13860

Offset: 1

Emeric Deutsch, Oct 15 2010

A 2-composition of n is a nonnegative matrix with two rows, such that each column has at least one nonzero entry and whose entries sum up to n.

			T(3,3) = 2 because we have (0/3) and (3/0) (the 2-compositions are written as (top row/bottom row)).
Triangle starts:
  2;
  5,2;
  12,10,2;
  29,41,10,2;
  70,152,46,10,2;

G. Castiglione, A. Frosini, E. Munarini, A. Restivo and S. Rinaldi, Combinatorial aspects of L-convex polyominoes, European J. Combin. 28 (2007), no. 6, 1724-1741.

Cf. A003480, A000129, A181339

h := proc (k) options operator, arrow: (1-z)^2/(1-4*z+2*z^2+2*z^(k+1)-z^(2*k+2)) end proc: f := proc (k) options operator, arrow; simplify(h(k)-h(k-1)) end proc: G := sum(f(k)*t^k, k = 1 .. 30): Gser := simplify(series(G, z = 0, 15)): for n to 11 do P[n] := sort(coeff(Gser, z, n)) end do: for n to 11 do seq(coeff(P[n], t, k), k = 1 .. n) end do; # yields sequence in triangular form

G.f. for 2-compositions with all entries <= k is h(k,z) = (1-z)^2/(1-4*z+2*z^2+2*z^(k+1)-z^(2*k+2)).
G.f. for 2-compositions with largest entry k is f(k,z) = h(k,z)-h(k-1,z) (these are the column g.f.'s).
G.f.: G(t,z) = Sum_{k>=1} f(k,z)*t^k.
Sum_{k=0..n} T(n,k) = A003480(n).
T(n,1) = A000129(n+1) (the Pell numbers).
Sum_{k=0..n} k*T(n,k) = A181339(n).

A181368 Triangle read by rows: T(n,k) is the number of L-convex polyominoes of semiperimeter n, having k maximal rectangles (n >= 2, 1 <= k <= floor(n/2)). An L-convex polyomino is a convex polyomino in which any two cells can be connected by a path internal to the polyomino and which has at most 1 change of direction (i.e., one of the four orientations of the letter L). A maximal rectangle in an L-convex polyomino P is a rectangle included in P that is maximal with respect to inclusion.

Original entry on oeis.org

1, 2, 3, 4, 4, 20, 5, 61, 16, 6, 146, 128, 7, 301, 584, 64, 8, 560, 1992, 704, 9, 966, 5641, 4272, 256, 10, 1572, 14002, 18880, 3584, 11, 2442, 31471, 67820, 27136, 1024, 12, 3652, 65428, 209820, 147200, 17408, 13, 5291, 127699, 579125, 640096, 157952

Offset: 2

Emeric Deutsch, Oct 17 2010

Row n contains floor(n/2) entries.
Sum of entries in row n is A003480(n-2).
Sum_{k>=1} k*T(n,k) = A181369(n).

			T(3,1)=2 because the L-convex polyominoes of semiperimeter 3 are the horizontal and the vertical dominoes, each containing one maximal rectangle.
Triangle starts:
  1;
  2;
  3,   4;
  4,  20;
  5,  61,  16;
  6, 146, 128;

G. Castiglione, A. Frosini, E. Munarini, A. Restivo and S. Rinaldi, Combinatorial aspects of L-convex polyominoes, European Journal of Combinatorics, 28, 2007, 1724-1741 (see Fig. 9).
G. Castiglione and A. Restivo, Reconstruction of L-convex polyominoes, Electronic Notes in Discrete Mathematics, Vol. 12, Elsevier Science, 2003.

Cf. A003480, A181369.

T := proc (n, k) options operator, arrow: sum(binomial(2*k-2, j)*binomial(n+j-1, 2*k+j-1), j = 0 .. 2*k-2) end proc: for n from 2 to 14 do seq(T(n, k), k = 1 .. floor((1/2)*n)) end do; # yields sequence in triangular form

T(n+2,k+1) = Sum_{j=0..2k} (-1)^j*2^(2k-j)*binomial(2k, j)*binomial(n+2k-j+1, 4k+1).
T(n+2,k+1) = Sum_{j=0..2k} binomial(2k, j)*binomial(n+j+1, 2k+j+1).
G.f. = G(t,z) = t*z^2*(1-z)^2/((1-z)^4 - t*z^2*(2-z)^2).

A253722 Triangle read by rows: coefficients of the partition polynomials for the reciprocal of the derivative of a power series, g(x)= 1/h'(x).

Original entry on oeis.org

1, -2, 4, -3, -8, 12, -4, 16, -36, 9, 16, -5, -32, 96, -54, -48, 24, 20, -6, 64, -240, 216, 128, -27, -144, -60, 16, 30, 24, -7, -128, 576, -720, -320, 216, 576, 160, -108, -96, -180, -72, 40, 36, 28, -8

Offset: 0

Tom Copeland, May 02 2015

This entry contains the integer coefficients of the partition polynomials P(n;h_1,h_2,...,h_(n+1)) for the reciprocal g(x) of the derivative of a power series in terms of the coefficients of the power series; i.e., g(x) = 1/[dh(x)/dx] = 1/[h_1 + 2*h_2 * x + 3*h_3 * x^2 + ...] = sum[n>=0, (h_1)^(-(n+1)) * P(n;h_1,...,h_(n+1)) * x^n].

This is a signed refinement of reversed A181289. See A145271, A133437, and A133314 for relations to compositional and multiplicative inversions.

			Let h(x) = h_0 + h_1 * x + h_2 * x^2 + ... . Then g(x) = 1/h'(x) = 1/[h_1 + 2*h_2 * x + 3*h_3 * x^2 + ...] = (h_1)^(-1) P(0;h_1) + (h_1)^(-2) * P(1;h_1,h_2) * x + (h_1)^(-3) * P(2;h_1,h_2,h_3) * x^2 + ... , and, with h_n = (n'), the first few partition polynomials are
P(0;..)=  1
P(1;..)= -2 (2')
P(2;..)=  4 (2')^2 - 3 (3')(1')
P(3;..)= -8 (2')^3 + 12 (3')(2')(1') - 4 (4')(1')^2
P(4;..)= 16 (2')^4 - 36 (2')^2(3')(1') + [9 (3')^2 + 16 (4')(2')](1')^2 - 5 (5')(1')^3
P(5;..)= -32 (2')^5 + 96 (2')^3(3')(1') + [-54 (3')^2(2') - 48 (4')(2')^2](1')^2 + [24 (3')(4') + 20 (5')(2')](1')^3 - 6 (6')(1')^4
P(6;..)= 64 (2')^6 - 240 (2')^4(3')(1') + [216 (3')^2(2') + 128 (4')(2')^3](1')^2 - [27 (3')^3 + 144 (4')(3')(2') + 60 (5')(2')^2](1')^3 + [16 (4')^2 + 30 (5')(3') + 24 (6')(2')](1')^4 - 7 (7')(1')^5

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Cf. A181289, A133437, A145271, A133314, A003480.

rows[n_] := {{1}}~Join~With[{s = 1/(1 + Sum[(k+1) u[k] x^k, {k, n}] + O[x]^(n+1))}, Table[Coefficient[s, x^k Product[u[t], {t, p}]], {k, n}, {p, Reverse@Sort[Sort /@ IntegerPartitions[k]]}]];
rows[7] // Flatten (* Andrey Zabolotskiy, Feb 19 2024 *)

C(v)={my(S=Set(v)); (-1)^(#v)*(#v)!*prod(i=1, #S, my(x=S[i], e=#select(y-> y==x, v)); (x+1)^e/e! )}
row(n)=[C(Vec(p)) | p<-Vecrev(partitions(n))]
{ for(n=0, 7, print(row(n))) } \\ Andrew Howroyd, Feb 19 2024

For the partition (1')^e(1)*(2')^e(2)*...*(n')^e(n) in P(m;...), the unsigned integer coefficient is [e(2)+e(3)+...+e(n)]! * [2^e(2)*3^e(3)*...*n^e(n)]/[e(2)!*e(3)!*...*e(n)!] with the sign determined by (-1)^[e(1) + m].
The partitions of P(m;..) are formed by adding one to each index of the partitions of m of Abramowitz and Stegun's partition table (p. 831; in the reversed order) and appending (1')^e(1) as a factor to obtain a partition of 2m.
Row sums are 1,-2,1,0,0,0,... . Row sums of the unsigned coefficients are A003480.

Row 7 added by Andrey Zabolotskiy, Feb 19 2024

A346432 a(0) = 1; a(n) = n! * Sum_{k=0..n-1} (n-k+1) * a(k) / k!.

Original entry on oeis.org

1, 2, 14, 144, 1968, 33600, 688320, 16450560, 449326080, 13806858240, 471395635200, 17703899136000, 725338710835200, 32193996432998400, 1538840509503897600, 78808952068374528000, 4305129487814098944000, 249876735246162984960000, 15356385691181506363392000

Offset: 0

Ilya Gutkovskiy, Jul 17 2021

nonn

Cf. A000670, A001339, A002866, A003480, A007840, A052555, A052567, A136658, A216794, A308939, A346433.

a[0] = 1; a[n_] := a[n] = n! Sum[(n - k + 1) a[k]/k!, {k, 0, n - 1}]; Table[a[n], {n, 0, 18}]
nmax = 18; CoefficientList[Series[1/(2 - 1/(1 - x)^2), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[(-1)^(n - k) StirlingS1[n, k] 2^k HurwitzLerchPhi[1/2, -k, 0]/2, {k, 0, n}], {n, 0, 18}]

my(x='x+O('x^25)); Vec(serlaplace(1 / (2 - 1 / (1 - x)^2))) \\ Michel Marcus, Jul 18 2021

E.g.f.: 1 / (2 - 1 / (1 - x)^2).
E.g.f.: 1 / (1 - Sum_{k>=1} (k+1) * x^k).
a(0) = 1, a(1) = 2, a(2) = 14; a(n) = 4 * n * a(n-1) - 2 * n * (n-1) * a(n-2).
a(n) = Sum_{k=0..n} (-1)^(n-k) * Stirling1(n,k) * 2^k * A000670(k).
a(n) = n! * A003480(n).

cp's OEIS Frontend

A181336 Triangle read by rows: T(n,k) is the number of 2-compositions of n having k even entries in the top row. A 2-composition of n is a nonnegative matrix with two rows, such that each column has at least one nonzero entry and whose entries sum up to n.

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A259860 a(n+8)+34*a(n+4)+a(n)=0 with a(0)-a(7) as shown.

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A259861 a(n+8)+34*a(n+4)+a(n)=0 with a(0)-a(7) as shown.

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A128611 Number of Z-convex polyominoes with semiperimeter n.

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A181293 Triangle read by rows: T(n,k) is the number of 2-compositions of n having k 0's (0<=k<=n) A 2-composition of n is a nonnegative matrix with two rows, such that each column has at least one nonzero entry and whose entries sum up to n.

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A181302 Triangle read by rows: T(n,k) is the number of 2-compositions of n having k columns with distinct entries (0<=k<=n).

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A181338 Triangle read by rows: T(n,k) is the number of 2-compositions of n having largest entry k (1<=k<=n).

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