A005773 -id:A005773 - cp's OEIS frontend

Previous Showing 41-50 of 103 results. Next

A062105 Square array read by antidiagonals: number of ways a pawn-like piece (with the initial 2-step move forbidden and starting from any square on the back rank) can end at various squares on an infinite chessboard.

1, 1, 2, 1, 3, 5, 1, 3, 8, 13, 1, 3, 9, 22, 35, 1, 3, 9, 26, 61, 96, 1, 3, 9, 27, 75, 171, 267, 1, 3, 9, 27, 80, 216, 483, 750, 1, 3, 9, 27, 81, 236, 623, 1373, 2123, 1, 3, 9, 27, 81, 242, 694, 1800, 3923, 6046, 1, 3, 9, 27, 81, 243, 721, 2038, 5211, 11257, 17303, 1, 3, 9, 27

Offset: 0

Antti Karttunen, May 30 2001

Table formatted as a square array shows the top-left corner of the infinite board.

The same array can also be constructed by the method used for constructing A217536, except with a top row consisting entirely of 1's instead of the natural numbers. - WG Zeist, Aug 25 2024

			Array begins:
 1       1       1       1       1       1       1       1       1       1       1
 2       3       3       3       3       3       3       3       3       3       3
 5       8       9       9       9       9       9       9       9       9 ...
 13      22      26      27      27      27      27      27      27 ...
 35      61      75      80      81      81      81 ...
 96      171     216     236     242     243 ...
 267     483     623     694     721 ...
 750     1373    1800    2038 ...
 2123    3923    5211 ...
 6046    11257 ...
 17303  ...
 ...
Formatted as a triangle:
 1,
 1, 2,
 1, 3, 5,
 1, 3, 8, 13,
 1, 3, 9, 22, 35,
 1, 3, 9, 26, 61, 96,
 1, 3, 9, 27, 75, 171, 267,
 1, 3, 9, 27, 80, 216, 483, 750,
 1, 3, 9, 27, 81, 236, 623, 1373, 2123,
 1, 3, 9, 27, 81, 242, 694, 1800, 3923, 6046,
 1, 3, 9, 27, 81, 243, 721, 2038, 5211, 11257, 17303,
 ...

Hans L. Bodlaender, The Chess Variant Pages
Hans L. Bodlaender et al., editors, The Piececlopedia (An overview of several fairy chess pieces)

A005773 gives the left column of the table. A000244 (powers of 3) gives the diagonal of the table. Variant of A062104. Cf. also A062103, A020474, A217536.

[seq(CPTVSeq(j),j=0..91)]; CPTVSeq := n -> ChessPawnTriangleV( (2+(n-((trinv(n)*(trinv(n)-1))/2))), ((((trinv(n)-1)*(((1/2)*trinv(n))+1))-n)+1) );
ChessPawnTriangleV := proc(r,c) option remember; if(r < 2) then RETURN(0); fi; if(c < 1) then RETURN(0); fi; if(2 = r) then RETURN(1); fi; RETURN(ChessPawnTriangleV(r-1,c-1)+ChessPawnTriangleV(r-1,c)+ChessPawnTriangleV(r-1,c+1)); end;
M:=12; T:=Array(0..M,0..M,0);
T[0,0]:=1; T[1,1]:=1;
for i from 1 to M do T[i,0]:=0; od:
for n from 2 to M do for k from 1 to n do
T[n,k]:= T[n,k-1]+T[n-1,k-1]+T[n-2,k-1];
od: od;
rh:=n->[seq(T[n,k],k=0..n)];
for n from 0 to M do lprint(rh(n)); od: # N. J. A. Sloane, Apr 11 2020

T[n_, k_] := T[n, k] = If[n < 1 || k < 1, 0, If[n == 1, 1, T[n - 1, k - 1] + T[n - 1, k] + T[n - 1, k + 1]]]; Table[T[n - k + 1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Mar 04 2016, adapted from PARI *)

T(n,k)=if(n<1 || k<1,0,if(n==1,1,T(n-1,k-1)+T(n-1,k)+T(n-1,k+1)))

Edited by N. J. A. Sloane, May 22 2014

A107230 A number triangle of inverse Chebyshev transforms.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 3, 6, 3, 1, 6, 12, 12, 4, 1, 10, 30, 30, 20, 5, 1, 20, 60, 90, 60, 30, 6, 1, 35, 140, 210, 210, 105, 42, 7, 1, 70, 280, 560, 560, 420, 168, 56, 8, 1, 126, 630, 1260, 1680, 1260, 756, 252, 72, 9, 1, 252, 1260, 3150, 4200, 4200, 2520, 1260, 360, 90, 10, 1

Offset: 0

Paul Barry, May 13 2005

First column is A001405, second column is A100071, third column is A107231. Row sums are A005773(n+1), diagonal sums are A026003. The inverse Chebyshev transform concerned takes a g.f. g(x)->(1/sqrt(1-4x^2))g(xc(x^2)) where c(x) is the g.f. of A000108. It transforms a(n) to b(n) = Sum_{k=0..floor(n/2)} binomial(n,k)*a(n-2k). Then a(n) = Sum_{k=0..floor(n/2)} (n/(n-k))*(-1)^k*binomial(n-k,k) *b(n-2k).

Triangle read by rows: T(n,k) is the number of paths of length n with steps U=(1,1), D=(1,-1) and H=(1,0), starting at (0,0), staying weakly above the x-axis (i.e., left factors of Motzkin paths) and having k H steps. Example: T(3,1)=6 because we have HUD. HUU, UDH, UHD, UHU and UUH. Sum_{k=0..n} k*T(n,k) = A132894(n). - Emeric Deutsch, Oct 07 2007

			Triangle begins
   1;
   1,  1;
   2,  2,  1;
   3,  6,  3,  1;
   6, 12, 12,  4,  1;
  10, 30, 30, 20,  5,  1;

Jinyuan Wang, Rows n=0..200 of triangle, flattened
Paul Barry, The Central Coefficients of a Family of Pascal-like Triangles and Colored Lattice Paths, J. Int. Seq., Vol. 22 (2019), Article 19.1.3.

Cf. A132894.

[[Binomial(n, k)*Binomial(n-k, Floor((n-k)/2)): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Feb 11 2019

T:=proc(n,k) options operator, arrow: binomial(n, k)*binomial(n-k, floor((1/2)*n-(1/2)*k)) end proc: for n from 0 to 11 do seq(T(n,k),k=0..n) end do; # yields sequence in triangular form - Emeric Deutsch, Oct 07 2007

Table[Binomial[n, k]*Binomial[n-k, Floor[(n-k)/2]], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Feb 11 2019 *)

T(n, k) = binomial(n, k)*binomial(n-k, (n-k)\2); \\ Michel Marcus, Feb 10 2019

[[binomial(n, k)*binomial(n-k, floor((n-k)/2)) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Feb 11 2019

T(n,k) = binomial(n,k)*binomial(n-k, floor((n-k)/2)).
G.f.: G=G(t,z) satisfies z*(1-2*z-t*z)*G^2+(1-2*z-t*z)*G-1=0. - Emeric Deutsch, Oct 07 2007
E.g.f.: exp(x*y)*(BesselI(0,2*x)+BesselI(1,2*x)). - Vladeta Jovovic, Dec 02 2008

A185100 Dihedral unlabeled Motzkin numbers: number of ways of drawing any number of nonintersecting chords joining n unlabeled points equally spaced on a circle, up to rotations and reflections of the circle.

Original entry on oeis.org

1, 1, 2, 2, 4, 5, 11, 16, 36, 65, 150, 312, 756, 1743, 4353, 10732, 27489, 70379, 183866, 481952, 1277784, 3402661, 9126689, 24584870, 66567924, 180939737, 493801694, 1352203202, 3715137460, 10237545525, 28291018283, 78384998904, 217715672036, 606103034821, 1691020991782, 4727601528674, 13242641322252, 37162431389051, 104469244613429

Offset: 0

Max Alekseyev, Feb 07 2011

nonn

Unlabeled version of A001006. Another version is given by A175954.

The number of ways of drawing exactly n chords joining 2n unlabeled points up to rotations and reflections is A006082(n+1). - Andrey Zabolotskiy, May 24 2018

Andrew Howroyd, Table of n, a(n) for n = 0..200
Andrew Howroyd, Chord Configuration Symmetries

Cf. A001006 (labeled points), A175954 (up to rotations only), A175955, A005773, A006082.

a1006[0] = 1; a1006[n_Integer] := a1006[n] = a1006[n - 1] + Sum[a1006[k]* a1006[n - 2 - k], {k, 0, n - 2}];
a142150[n_] := n*(1 + (-1)^n)/4;
a2426[n_] := Coefficient[(1 + x + x^2)^n, x, n];
a175954[0] = 1; a175954[n_] := (1/n)*(a1006[n] + a142150[n]*a1006[n/2 - 1] + Sum[EulerPhi[n/d]*a2426[d], {d, Most @Divisors[n]}]);
a5773[0] = 1; a5773[n_] := Sum[k/n*Sum[Binomial[n, j]*Binomial[j, 2*j - n - k], {j, 0, n}], {k, 1, n}];
a[0] = 1;
a[n_?OddQ] := With[{m = (n-1)/2}, (1/2)*(a175954[2*m + 1] + a5773[m + 1])];
a[n_?EvenQ] := With[{m = n/2}, (1/4)*(2*a175954[2*m] + a5773[m] + a5773[m + 1] + a1006[m - 1])];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Jul 02 2018, after Andrew Howroyd *)

a(2n+1) = (1/2) * (A175954(2n+1) + A005773(n+1)). - Andrew Howroyd, Apr 01 2017

a(2n) = (1/4) * (2 * A175954(2n) + A005773(n) + A005773(n+1) + A001006(n-1)) for n > 0. - Andrew Howroyd, Apr 01 2017

A264868 Number of rooted tandem duplication trees on n gene segments.

Original entry on oeis.org

1, 1, 2, 6, 22, 92, 420, 2042, 10404, 54954, 298648, 1660714, 9410772, 54174212, 316038060, 1864781388, 11111804604, 66782160002, 404392312896, 2465100947836, 15116060536540, 93184874448186, 577198134479356, 3590697904513792, 22425154536754776

Offset: 1

Peter Bala, Nov 27 2015

Apparently a(n) is the number of words [d(0)d(1)d(2)...d(n)] where d(k) <= k (so d(0)=0) and if w(k-1) > w(k) then w(k-1) - w(k) = 1 (that is, descents by 2 or more are forbidden). - Joerg Arndt, Jan 26 2024

			Form _Joerg Arndt_, Jan 26 2024: (Start)
The a(5) = 22 words as described in the comment are (dots denote zeros, leading zeros omitted):
    1:  [ . . . ]
    2:  [ . . 1 ]
    3:  [ . . 2 ]
    4:  [ . . 3 ]
    5:  [ . 1 . ]
    6:  [ . 1 1 ]
    7:  [ . 1 2 ]
    8:  [ . 1 3 ]
    9:  [ . 2 1 ]
   10:  [ . 2 2 ]
   11:  [ . 2 3 ]
   12:  [ 1 . . ]
   13:  [ 1 . 1 ]
   14:  [ 1 . 2 ]
   15:  [ 1 . 3 ]
   16:  [ 1 1 . ]
   17:  [ 1 1 1 ]
   18:  [ 1 1 2 ]
   19:  [ 1 1 3 ]
   20:  [ 1 2 1 ]
   21:  [ 1 2 2 ]
   22:  [ 1 2 3 ]
(End)

Mathematics of Evolution and Phylogeny, O. Gascuel (ed.), Oxford University Press, 2005

Alois P. Heinz, Table of n, a(n) for n = 1..1000
O. Gascuel, M. Hendy, A. Jean-Marie and R. McLachlan, The combinatorics of tandem duplication trees, Systematic Biology 52, (2003), 110-118.
J. Yang and L. Zhang, Letter. On Counting Tandem Duplication Trees, Molecular Biology and Evolution, Volume 21, Issue 6, (2004) 1160-1163.

Cf. A086521, A264869, A264870, A291680.

Cf. A005773.

a:= proc(n) option remember;
       if n = 1 then 1 elif n = 2 then 1 else add((-1)^(k+1)*
          binomial(n+1-2*k, k)*a(n-k), k = 1..floor((n+1)/3))
       end if;
    end proc:
seq(a(n), n = 1..24);

a[n_] := a[n] = If[n == 1, 1, If[n == 2, 1, Sum[(-1)^(k+1) Binomial[n+1-2k, k] a[n-k], {k, 1, Floor[(n+1)/3]}]]]; Array[a, 25] (* Jean-François Alcover, May 29 2019 *)

from sympy.core.cache import cacheit
from sympy import binomial
@cacheit
def a(n):
    return 1 if n<3 else sum([(-1)**(k + 1)*binomial(n + 1 - 2*k, k)*a(n - k) for k in range(1, (n + 1)//3 + 1)])
print([a(n) for n in range(1, 26)]) # Indranil Ghosh, Aug 30 2017

a(n) = Sum_{k = 1..floor((n + 1)/3)} (-1)^(k + 1)*binomial(n + 1 - 2*k,k)*a(n-k) with a(1) = a(2) = 1 (Yang and Zhang).
For n >= 3, (1/2)*a(n) = A086521(n) is the number of tandem duplication trees on n gene segments.
Main diagonal and row sums of A264869.
a(n) = Sum_{k=0..n-1} A291680(n-1,k). - Alois P. Heinz, Aug 29 2017

A026135 Number of (s(0),s(1),...,s(n)) such that every s(i) is a nonnegative integer, s(0) = 1, |s(1) - s(0)| = 1, |s(i) - s(i-1)| <= 1 for i >= 2. Also sum of numbers in row n+1 of the array T defined in A026120.

Original entry on oeis.org

1, 2, 5, 14, 39, 110, 312, 890, 2550, 7334, 21161, 61226, 177575, 516114, 1502867, 4383462, 12804429, 37452870, 109682319, 321563658, 943701141, 2772060618, 8149661730, 23978203662, 70600640796, 208014215066, 613266903927

Offset: 0

Clark Kimberling

nonn

a(n) is the total number of rows of consecutive peaks in all Motzkin (n+2)-paths. For example, with U=upstep, D=downstep, F=flatstep, the path FU(UD)FU(UDUDUD)DD(UD) contains 3 rows of peaks (in parentheses). The 9 Motzkin 4-paths are FFFF, FF(UD), F(UD)F, FUFD, (UD)FF, (UDUD), UFDF, UFFD, U(UD)D, containing a total of 5 rows of peaks and so a(2)=5. - David Callan, Aug 16 2006

G. C. Greubel, Table of n, a(n) for n = 0..1000

First differences are in A025566, second differences in A005773.

Pairwise sums of A025179.

CoefficientList[Series[((x - 1)^2*((1 + x)/(1 - 3 x))^(1/2) + x^2 - 1)/(2*x^2), {x,0,50}], x] (* G. C. Greubel, May 22 2017 *)

x='x+O('x^50); Vec(((x-1)^2*((1+x)/(1-3x))^(1/2) + x^2 - 1)/(2*x^2)) \\ G. C. Greubel, May 22 2017

a(n) = Sum_{k=0..n} binomial(n-1, k-1)*binomial(k+1, floor((k+1)/2)). - Vladeta Jovovic, Sep 18 2003
G.f.: ((x-1)^2*((1+x)/(1-3x))^(1/2) + x^2 - 1)/(2*x^2). - David Callan, Aug 16 2006
G.f. = (1+z)*(1+z^2)/(1-z) where z=x*A001006(x). [From R. J. Mathar, Jul 07 2009]
Conjecture: (n+2)*a(n) +3*(-n-1)*a(n-1) +(-n-2)*a(n-2) +3*(n-3)*a(n-3)=0. - R. J. Mathar, Jun 23 2013

More terms from David Callan, Aug 16 2006

Typo in a(19) corrected by R. J. Mathar, Jul 07 2009

A054392 Number of permutations with certain forbidden subsequences.

Original entry on oeis.org

1, 1, 2, 5, 14, 42, 131, 418, 1352, 4410, 14463, 47605, 157084, 519255, 1718653, 5693903, 18877509, 62620857, 207816230, 689899944, 2290913666, 7608939443, 25276349558, 83977959853, 279039638062, 927272169336, 3081641953082

Offset: 0

N. J. A. Sloane, Elisa Pergola (elisa(AT)dsi.unifi.it), May 21 2000

nonn

Apparently the Motzkin transform of A005251, after A005251(0) is set to 1. - R. J. Mathar, Dec 11 2008

			G.f. = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 131*x^6 + 418*x^7 + 1352*x^8 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
E. Barcucci et al., From Motzkin to Catalan Permutations, Discr. Math., 217 (2000), 33-49.
Nickolas Hein, Jia Huang, Variations of the Catalan numbers from some nonassociative binary operations, arXiv:1807.04623 [math.CO], 2018.

Interpolates between Motzkin numbers (A001006) and Catalan numbers (A000108).

Cf. A005773, A054391-A054394.

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (2 -10*x +13*x^2 -5*x^3 +x^2*sqrt(1-2*x-3*x^2))/(2-12*x+22*x^2-14*x^3) )); // G. C. Greubel, Feb 14 2020

m:=30; S:=series((2-10*x+13*x^2-5*x^3+x^2*sqrt(1-2*x-3*x^2))/(2-12*x+22*x^2 -14*x^3), x, m+1): seq(coeff(S, x, j), j=0..m); # G. C. Greubel, Feb 14 2020

a[0] = 1; a[n_]:= Module[{M}, M = Table[If[jJean-François Alcover, Aug 16 2018, after A054391 *)
a[n_]:= a[n]= If[n<2, 1, If[n==2, 2, If[3<=n<=4, 9*n-22, ((8*n-19)*a[n-1] - (20*n-49)*a[n-2] +(11*n-1)*a[n-3] +(19*n-116)*a[n-4] -21*(n-5)*a[n-5])/(n-2) ]]]; Table[a[n], {n,0,30}] (* G. C. Greubel, Feb 14 2020 *)

PARI

{a(n) = if( n<1, n==0, polcoeff( subst( x * (1 - x) / (1 - 2*x + x^2 - x^3), x, serreverse( x / (1 + x + x^2) + x * O(x^n))), n))}; /* Michael Somos, Aug 06 2014 */

Sage

def A054392_list(prec): P. = PowerSeriesRing(ZZ, prec) return P( (2-10*x+13*x^2-5*x^3+x^2*sqrt(1-2*x-3*x^2))/(2-12*x+22*x^2-14*x^3) ).list() A054392_list(30) # G. C. Greubel, Feb 14 2020

Formula

(n-2)*a(n) = (8*n-19)*a(n-1) - (20*n-49)*a(n-2) + (11*n-1)*a(n-3) + (19*n-116) * a(n-4) - 21*(n-5)*a(n-5). - R. J. Mathar, Aug 09 2015

G.f.: (2 -10*x +13*x^2 -5*x^3 +x^2*sqrt(1-2*x-3*x^2))/(2-12*x+22*x^2-14*x^3). - Michael D. Weiner, Feb 07 2020

A066822 The fourth column of A038622, triangular array that counts rooted polyominoes.

Original entry on oeis.org

1, 5, 20, 71, 238, 770, 2436, 7590, 23397, 71566, 217646, 659022, 1988805, 5986176, 17980968, 53922096, 161492571, 483149385, 1444245936, 4314214443, 12880107548, 38436170366, 114657076900, 341926185770, 1019435748435, 3038815305981, 9056974493700

Offset: 0

Views

Graph

List

Refs

Listen

History

Text

Internal format

Author

Randall L Rathbun, Jan 19 2002

Keywords

easy
nice
nonn

Comments

There is a general solution for all rows of this triangular array: For the k-th row and n-th term on this row: a(0)=0; a(1)=1; a(n) = (2*k-1+n)*n*a(n) = 2*(n+k)*(n+k-1)*a(n-1) + 3*(n+k-1)*(n+k-2)*a(n-2).

Links

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

D. Gouyou-Beauchamps and G. Viennot, Equivalence of the two-dimensional directed animal problem to a one-dimensional path problem, Adv. in Appl. Math. 9 (1988), no. 3, 334-357.

Crossrefs

Cf. A038622.

Cf. A005773, A005774, A005775.

Programs

Haskell

a066822 = flip a038622 3 . (+ 3) -- Reinhard Zumkeller, Feb 26 2013

Maple

a := n -> simplify(GegenbauerC(n,-n+1-4,-1/2)+GegenbauerC(n-1,-n-3,-1/2)): seq(a(n), n=0..20); # Peter Luschny, May 12 2016

Mathematica

Table[GegenbauerC[n,-n-3,-1/2]+GegenbauerC[n-1,-n-3,-1/2],{n,0,40}] (* Harvey P. Dale, Feb 20 2017 *)

PARI

s=[0,1]; {A038622(n,k)=if(n==0,1,t=(2*(n+k)*(n+k-1)*s[2]+3*(n+k-1)*(n+k-2)*s[1])/((n+2*k-1)*n); s[1]=s[2]; s[2]=t; t)}

Formula

a(0)=0; a(1)=1; (n+7)*n*a(n)=2*(n+4)*(n+3)*a(n-1) + 3*(n+3)*(n+2)*a(n-2).

a(n) = ((-3)^(1/2)/9)*(-2*(n+7)^(-1)*(n+4)*(-1)^n*hypergeom([3/2, n+6],[2],4/3)-(n+6)^(-1)*(-1)^n*(5*n+18)*hypergeom([3/2, n+5],[2],4/3)). - Mark van Hoeij, Oct 31 2011

a(n) ~ 3^(n+7/2) / sqrt(Pi*n). - Vaclav Kotesovec, Mar 10 2014

a(n) = GegenbauerC(n,-n+1-4,-1/2)+GegenbauerC(n-1,-n-3,-1/2). - Peter Luschny, May 12 2016

Extensions

More terms from Harvey P. Dale, Feb 20 2017

A081113 Number of paths of length n-1 a king can take from one side of an n X n chessboard to the opposite side.

Original entry on oeis.org

1, 4, 17, 68, 259, 950, 3387, 11814, 40503, 136946, 457795, 1515926, 4979777, 16246924, 52694573, 170028792, 546148863, 1747255194, 5569898331, 17698806798, 56076828573, 177208108824, 558658899825, 1757365514652

Offset: 1

Views

Graph

List

Refs

Listen

History

Text

Internal format

Author

David Scambler, Apr 16 2003

Keywords

easy
nonn

Comments

a(n) = number of sequences (a_1,a_2,...,a_n) with 1<=a_i<=n for all i and |a_(i+1)-a_(i)|<=1 for 1<=i<=n-1. For n=2 the sequences are 11, 12, 21, 22. - David Callan, Oct 24 2004

Simon Plouffe proposes the ordinary generating function A(x) (for offset zero) in the implicit form 3-10*x+12*x^2+(-4+30*x+54*x^3-72*x^2)*A(x)+(81*x^4+54*x^2+1-12*x-108*x^3)*A(x)^2 = 0 which delivers at least the first 200 terms (i.e., as far as tested) correctly. - David Scambler, R. J. Mathar, Jan 06 2011

Examples

For n=2 the 4 paths are (0,0)->(0,1); (0,0)->(1,1); (1,0)->(0,1); (1,0)->(1,1).

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Simon Plouffe, OEIS conjectured formulas.

D. Yaqubi, M. Farrokhi D. G., and H. Ghasemian Zoeram, Lattice paths inside a table, I , arXiv:1612.08697 [math.CO], 2016-2017.

Crossrefs

Cf. A005773 (paths which begin at a corner), diagonal of A296449.

Programs

Maple

A026300 := proc(n,k) add( binomial(n,2*i+n-k)*(binomial(2*i+n-k,i) -binomial(2*i+n-k,i-1)), i=0..floor(k/2)) ; end proc: A081113 := proc(n) add(k*(n-k+1)*A026300(n-1,k-1),k=1..n) ; end proc: seq(A081113(n),n=1..20) ; # R. J. Mathar, Jun 09 2010

Mathematica

t[n_, k_] := Sum[ Binomial[n, 2i + n - k] (Binomial[2i + n - k, i] - Binomial[2i + n - k, i - 1]), {i, 0, Floor[k/2]}] (* from A026300 *); f[n_] := Sum[ k(n - k + 1)t[n - 1, k - 1], {k, n}]; Array[f, 24]

Formula

a(n) = Sum_{k=1..n} k*(n-k+1)*M(n-1, k-1) where k*(n-k+1) is the triangular view of A003991 and M() is the Motzkin triangle A026300.

Conjecture: g.f.(x)=z*A064808(z), where z=x*A001006(x) and A...(x) are the corresponding generating functions. - R. J. Mathar, Jul 07 2009

Conjecture from WolframAlpha (verified for 1<=n<=180): (n+3)*a(n+4) = 27*n*a(n) +27*a(n+1) -9*(2*n+5)*a(n+2) +(8*n+21)*a(n+3). - Alexander R. Povolotsky, Jan 04 2011

Shorter recurrence: (n-1)*(2*n-7)*a(n) = (10*n^2-39*n+23)*a(n-1) - 3*(2*n^2-n-9)*a(n-2) - 9*(n-3)*(2*n-5)*a(n-3). - Vaclav Kotesovec, Oct 28 2012

a(n) ~ 3^(n-1)*n*(1-4/(sqrt(3*Pi*n))). - Vaclav Kotesovec, Oct 28 2012

a(n) = (n+2)*3^(n-2)+2*Sum_{k=0..n-3} (n-k-2)*3^(n-k-3)*A001006(k). [Yaqubi Corollary 2.8] - R. J. Mathar, Dec 13 2017

A115990 Riordan array (1/sqrt(1-2*x-3*x^2), (1-2*x-3*x^2)/(2*(1-3*x)) - sqrt(1-2*x-3*x^2)/2).

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 7, 5, 3, 1, 19, 13, 8, 4, 1, 51, 35, 22, 12, 5, 1, 141, 96, 61, 35, 17, 6, 1, 393, 267, 171, 101, 53, 23, 7, 1, 1107, 750, 483, 291, 160, 77, 30, 8, 1, 3139, 2123, 1373, 839, 476, 244, 108, 38, 9, 1, 8953, 6046, 3923, 2423, 1406, 752, 360, 147, 47, 10

Offset: 0

Views

Graph

List

Refs

Listen

History

Text

Internal format

Author

Paul Barry, Feb 10 2006

Keywords

easy
nonn
tabl

Comments

First column is central trinomial coefficients A002426. Second column is number of directed animals of size n+1, A005773(n+1). Row sums are A005717 (number of horizontal steps in all Motzkin paths of length n). First column has e.g.f. exp(x) I_0(2x). Row sums have e.g.f. dif(exp(x) I_1(2x),x).

Riordan array (1/sqrt(1-2*x-3*x^2), (1+x-sqrt(1-2*x-3*x^2))/2).

Examples

Triangle begins 1; 1, 1; 3, 2, 1; 7, 5, 3, 1; 19, 13, 8, 4, 1; 51, 35, 22, 12, 5, 1; 141, 96, 61, 35, 17, 6, 1;

Links

G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened

Crossrefs

Cf. A115991, A005773 (k=1), A025566 (k=2), A035045 (k=3), A152948 (diag. n=k+2), .

Programs

GAP

Flat(List([0..10], n-> List([0..n], k-> Sum([0..n], j-> Binomial(n-k, j-k)*Binomial(j, n-j)) ))); # G. C. Greubel, May 09 2019

Magma

[[(&+[Binomial(n-k, j-k)*Binomial(j, n-j): j in [0..n]]): k in [0..n]]: n in [0..10]]; // G. C. Greubel, May 09 2019

Maple

A115990 := proc(n,k) add(binomial(n-k,j-k)*binomial(j,n-j),j=0..n) ; end proc: seq(seq(A115990(n,k),k=0..n),n=0..12) ; # R. J. Mathar, Jun 25 2023

Mathematica

Table[Sum[ Binomial[n-k, j-k]*Binomial[j, n-j], {j, 0, n}], {n, 0, 10}, {k, 0, n} ] // Flatten (* G. C. Greubel, Mar 07 2017 *)

PARI

{T(n, k) = sum(j=0, n, binomial(n-k, j-k)*binomial(j, n-j))}; \\ G. C. Greubel, May 09 2019

Sage

[[sum(binomial(n-k, j-k)*binomial(j, n-j) for j in (0..n)) for k in (0..n)] for n in (0..10)] # G. C. Greubel, May 09 2019

Formula

Number triangle T(n,k) = Sum_{j=0..n} C(n-k,j-k)*C(j,n-j).

A158793 Triangle read by rows: product of A130595 and A092392 considered as infinite lower triangular arrays.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 7, 4, 1, 1, 19, 9, 5, 1, 1, 51, 26, 11, 6, 1, 1, 141, 70, 34, 13, 7, 1, 1, 393, 197, 92, 43, 15, 8, 1, 1, 1107, 553, 265, 117, 53, 17, 9, 1, 1, 3139, 1570, 751, 346, 145, 64, 19, 10, 1, 1, 8953, 4476, 2156, 991, 441, 176, 76, 21, 11, 1, 1

Offset: 0

Views

Graph

List

Refs

Listen

History

Text

Internal format

Author

Gary W. Adamson & Roger L. Bagula, Mar 26 2009

Keywords

nonn
tabl

Comments

Riordan array (f(x), x*g(x)) where f(x) is the g.f. of A002426 and where g(x) is the g.f. of A005043. - Philippe Deléham, Dec 05 2009

Matrix product P * Q * P^(-1), where P denotes Pascal's triangle A007318 and Q denotes A061554 (formed from P by sorting the rows into descending order). Cf. A158815 and A171243. - Peter Bala, Jul 13 2021

Examples

First rows of the triangle: 1; 1, 1; 3, 1, 1; 7, 4, 1, 1; 19, 9, 5, 1, 1; 51, 26, 11, 6, 1, 1; 141, 70, 34, 13, 7, 1, 1; 393, 197, 92, 43, 15, 8, 1, 1; 1107, 553, 265, 117, 53, 17, 9, 1, 1; 3139, 1570, 751, 346, 145, 64, 19, 10, 1, 1; 8953, 4476, 2156, 991, 441, 176, 76, 21, 11, 1, 1;

Crossrefs

T(n, 0) = A002426(n), A005773 (row sums).

Cf. A046899, A007318, A158815, A171243.

Programs

Maple

A158793 := proc (n, k) add((-1)^(n+j)*binomial(n, j)*binomial(2*j-k, j-k), j = k..n); end proc: seq(seq(A158793(n, k), k = 0..n), n = 0..10); # Peter Bala, Jul 13 2021

Mathematica

T[n_, k_] := (-1)^(k + n) Binomial[n, k] HypergeometricPFQ[{k/2 + 1/2, k/2 + 1, k - n}, {k + 1, k + 1}, 4]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Peter Luschny, Jul 17 2021 *)

Formula

T(n, m) = Sum_{k=m..n-1} A130595(n,k) * A092392(k+1,m+1), with the triangular interpretation of A092392.

Conjecture: T(n,1) = A113682(n-1). - R. J. Mathar, Oct 06 2009

Sum_{k=0..n} T(n,k)*x^k = A002426(n), A005773(n+1), A000244(n), A126932(n) for x = 0,1,2,3 respectively. - Philippe Deléham, Dec 03 2009

T(n, k) = (-1)^(k + n) binomial(n, k) hypergeom([k/2 + 1/2, k/2 + 1, k - n], [k + 1, k + 1], 4). - Peter Luschny, Jul 17 2021

Extensions

Simplified definition from R. J. Mathar, Oct 06 2009

Previous Showing 41-50 of 103 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.
Content is available under The OEIS End-User License Agreement.

cp's OEIS Frontend

A062105 Square array read by antidiagonals: number of ways a pawn-like piece (with the initial 2-step move forbidden and starting from any square on the back rank) can end at various squares on an infinite chessboard.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Extensions

A107230 A number triangle of inverse Chebyshev transforms.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Sage

Formula

A185100 Dihedral unlabeled Motzkin numbers: number of ways of drawing any number of nonintersecting chords joining n unlabeled points equally spaced on a circle, up to rotations and reflections of the circle.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A264868 Number of rooted tandem duplication trees on n gene segments.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Python

Formula

A026135 Number of (s(0),s(1),...,s(n)) such that every s(i) is a nonnegative integer, s(0) = 1, |s(1) - s(0)| = 1, |s(i) - s(i-1)| <= 1 for i >= 2. Also sum of numbers in row n+1 of the array T defined in A026120.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A054392 Number of permutations with certain forbidden subsequences.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Sage

Formula

A115990 Riordan array (1/sqrt(1-2x-3x^2), (1-2x-3x^2)/(2(1-3x)) - sqrt(1-2x-3x^2)/2).