A005043 -id:A005043 - cp's OEIS frontend

A032027 Number of planted planar trees (n+1 nodes) where any 2 subtrees extending from the same node are different.

1, 1, 1, 3, 5, 13, 35, 95, 255, 715, 2081, 6003, 17645, 52127, 155863, 468129, 1415521, 4301055, 13134789, 40275109, 123970669, 382919917, 1186475687, 3686899725, 11487023793, 35876838669, 112304155021, 352276801491

Offset: 1

Christian G. Bower

			From _Gus Wiseman_, Nov 15 2022: (Start)
The a(1) = 1 through a(6) = 13 ordered rooted identity trees (ranked by A358374):
  o  (o)  ((o))  ((o)o)   (((o))o)   (((o)o)o)
                 (o(o))   (((o)o))   ((o(o))o)
                 (((o)))  ((o(o)))   (o((o)o))
                          (o((o)))   (o(o(o)))
                          ((((o))))  ((((o)))o)
                                     ((((o))o))
                                     ((((o)o)))
                                     (((o))(o))
                                     (((o(o))))
                                     ((o)((o)))
                                     ((o((o))))
                                     (o(((o))))
                                     (((((o)))))
(End)

Andrew Howroyd, Table of n, a(n) for n = 1..200
C. G. Bower, Transforms (2)
Index entries for sequences related to rooted trees

The unordered version is A004111, ranked by A276625.
These trees (ordered rooted identity) are ranked by A358374.
Cf. A000081, A001190, A005043, A003238, A063895, A358377.

aot[n_]:=If[n==1,{{}},Join@@Table[Tuples[aot/@c],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[Select[aot[n],FreeQ[#,[_]?(!UnsameQ@@#&)]&]],{n,1,10}] (* Gus Wiseman, Nov 15 2022 *)

AGK(v)={apply(p->subst(serlaplace(y^0*p),y,1), Vec(prod(k=1, #v, (1 + x^k*y + O(x*x^#v))^v[k])-1, -#v))}
seq(n)={my(v=[1]); for(i=2, n, v=concat([1], AGK(v))); v} \\ Andrew Howroyd, Sep 20 2018

Shifts left under "AGK" (ordered, elements, unlabeled) transform.

A099323 Expansion of (sqrt(1+3x) + sqrt(1-x))/(2sqrt(1-x)).

Original entry on oeis.org

1, 1, 0, 1, -1, 3, -6, 15, -36, 91, -232, 603, -1585, 4213, -11298, 30537, -83097, 227475, -625992, 1730787, -4805595, 13393689, -37458330, 105089229, -295673994, 834086421, -2358641376, 6684761125, -18985057351, 54022715451, -154000562758, 439742222071, -1257643249140

Offset: 0

Paul Barry, Oct 12 2004

Binomial transform is A072100.
Signed Motzkin numbers with an additional leading 1.
Inverse binomial transform of A001405 gives this without the initial 1. So does the binomial transform of (-1)^n*A000108(n) = [1,-1,2,-5,14,-42,...]. - Philippe Deléham, Mar 20 2007

G. C. Greubel, Table of n, a(n) for n = 0..1000
C. Banderier and D. Merlini, Lattice paths with an infinite set of jumps, FPSAC'02 Melbourne, 2002.

Cf. A000108, A001405, A005043, A072100.

A099323:= func< n | (&+[(-1)^k*Binomial(n-1, k)*Catalan(k): k in [0..n]]) >;
[A099323(n): n in [0..40]]; // G. C. Greubel, Nov 25 2021

with(PolynomialTools): CoefficientList(convert(taylor((sqrt(1 + 3*x) + sqrt(1 - x))/2/sqrt(1 - x), x = 0, 33), polynom), x); # Taras Goy, Aug 07 2017

CoefficientList[Series[(Sqrt[1+3x]+Sqrt[1-x])/(2Sqrt[1-x]),{x,0,40}],x] (* Harvey P. Dale, Feb 06 2015 *)

[sum((-1)^k*binomial(n-1, k)*catalan_number(k) for k in (0..n)) for n in (0..40)] # G. C. Greubel, Nov 25 2021

a(n) = 0^n + Sum_{k=0..n-1} binomial(n-1,k)*(-1)^k*C(k), where C(k) is the k-th Catalan number.
G.f.: 1 + x/(1-sqrt(x))/G(0), where G(k)= 1 +  sqrt(x)/(1  -  sqrt(x)/(1 + x/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jul 28 2013
D-finite with recurrence: n*a(n) + 2*(n-2)*a(n-1) + 3*(-n+2)*a(n-2) = 0. - R. J. Mathar, Oct 10 2014
a(n) ~ -(-1)^n * 3^(n + 1/2) / (8*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 31 2017

Edited by N. J. A. Sloane, Oct 05 2009

A358379 Edge-height (or depth) of the n-th standard ordered rooted tree.

Original entry on oeis.org

0, 1, 2, 1, 3, 2, 2, 1, 2, 3, 2, 2, 3, 2, 2, 1, 4, 2, 3, 3, 3, 2, 2, 2, 2, 3, 2, 2, 3, 2, 2, 1, 3, 4, 2, 2, 3, 3, 3, 3, 2, 3, 2, 2, 3, 2, 2, 2, 4, 2, 3, 3, 3, 2, 2, 2, 2, 3, 2, 2, 3, 2, 2, 1, 3, 3, 4, 4, 3, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 4, 2, 3, 3, 3, 2, 2

Offset: 1

Gus Wiseman, Nov 16 2022

nonn

We define the n-th standard ordered rooted tree to be obtained by taking the (n-1)-th composition in standard order (graded reverse-lexicographic, A066099) as root and replacing each part with its own standard ordered rooted tree. This ranking is an ordered variation of Matula-Goebel numbers, giving a bijective correspondence between positive integers and unlabeled ordered rooted trees.

			The standard ordered rooted tree ranking begins:
  1: o        10: (((o))o)   19: (((o))(o))
  2: (o)      11: ((o)(o))   20: (((o))oo)
  3: ((o))    12: ((o)oo)    21: ((o)((o)))
  4: (oo)     13: (o((o)))   22: ((o)(o)o)
  5: (((o)))  14: (o(o)o)    23: ((o)o(o))
  6: ((o)o)   15: (oo(o))    24: ((o)ooo)
  7: (o(o))   16: (oooo)     25: (o(oo))
  8: (ooo)    17: ((((o))))  26: (o((o))o)
  9: ((oo))   18: ((oo)o)    27: (o(o)(o))
For example, the 52nd ordered tree is (o((o))oo) so a(52) = 3.

Gus Wiseman, Statistics, classes, and transformations of standard compositions

Records occur at A004249.
The triangle counting trees by this statistic is A080936, unordered A034781.
Unordered version is A109082, nodes A061775, leaves A109129, edges A196050.
Leaves are counted by A358371.
Nodes are counted by A358372.
Node-height is a(n) + 1, unordered version is A358552.
A000081 counts unordered rooted trees, ranked by A358378.
A000108 counts ordered rooted trees.
A001263 counts ordered rooted trees by leaves.
Cf. A005043, A055277, A187306, A358373, A358374, A358375, A358376, A358377.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
srt[n_]:=If[n==1,{},srt/@stc[n-1]];
Table[Depth[srt[n]]-2,{n,100}]

A279944 Number of positions in the free pure symmetric multifunction in one symbol with j-number n.

Original entry on oeis.org

1, 3, 5, 5, 7, 7, 9, 4, 7, 9, 11, 6, 9, 11, 13, 7, 8, 11, 13, 15, 9, 10, 13, 15, 9, 17, 6, 11, 12, 15, 17, 6, 11, 19, 8, 9, 13, 14, 17, 19, 8, 13, 21, 10, 11, 15, 16, 19, 11, 21, 10, 15, 23, 12, 13, 17, 18, 21, 13, 23, 12, 17, 25, 7, 14, 15, 19, 20, 23, 15, 25, 14, 19, 27, 9, 16, 17, 21, 22, 25, 9, 17, 27, 16, 21, 29, 11, 18, 19, 23, 24, 27, 11, 19, 29, 18, 23, 31, 13, 11

Offset: 1

Gus Wiseman, Dec 24 2016

nonn

A free pure symmetric multifunction in one symbol f in PSM(x) is either (case 1) f = the symbol x, or (case 2) f = an expression of the form h[g_1,...,g_k] where h is in PSM(x), each of the g_i for i=1..(k>0) is in PSM(x), and for i < j we have g_i <= g_j under a canonical total ordering of PSM(x), such as the Mathematica ordering of expressions. For a positive integer n we define a free pure symmetric multifunction j(n) by: j(1)=x; j(n>1) = j(h)[j(g_1),...,j(g_k)] where n = r(h)^(p(g_1)*...*p(g_k)-1). Here r(n) is the n-th number that is not a perfect power (A007916) and p(n) is the n-th prime number (A000040). See example. Then a(n) is the number of brackets [...] plus the number of x's in j(n).

			The first 20 free pure symmetric multifunctions in x are:
j(1)  = j(1)            = x
j(2)  = j(1)[j(1)]      = x[x]
j(3)  = j(2)[j(1)]      = x[x][x]
j(4)  = j(1)[j(2)]      = x[x[x]]
j(5)  = j(3)[j(1)]      = x[x][x][x]
j(6)  = j(4)[j(1)]      = x[x[x]][x]
j(7)  = j(5)[j(1)]      = x[x][x][x][x]
j(8)  = j(1)[j(1),j(1)] = x[x,x]
j(9)  = j(2)[j(2)]      = x[x][x[x]]
j(10) = j(6)[j(1)]      = x[x[x]][x][x]
j(11) = j(7)[j(1)]      = x[x][x][x][x][x]
j(12) = j(8)[j(1)]      = x[x,x][x]
j(13) = j(9)[j(1)]      = x[x][x[x]][x]
j(14) = j(10)[j(1)]     = x[x[x]][x][x][x]
j(15) = j(11)[j(1)]     = x[x][x][x][x][x][x]
j(16) = j(1)[j(3)]      = x[x[x][x]]
j(17) = j(12)[j(1)]     = x[x,x][x][x]
j(18) = j(13)[j(1)]     = x[x][x[x]][x][x]
j(19) = j(14)[j(1)]     = x[x[x]][x][x][x][x]
j(20) = j(15)[j(1)]     = x[x][x][x][x][x][x][x].

Cf. A005043, A007916, A106490, A277564, A277615, A277996, A278028, A280000.

Cf. A279984 (numbers j(n)[x]=j(prime(n))), A277576 (numbers j(n)=x[x][x][x]...), A058891 (numbers j(n)=x[x,...,x]), A279969 (numbers j(n)=x[x[...[x]]]).

nn=100;
radQ[n_]:=If[n===1,False,SameQ[GCD@@FactorInteger[n][[All,2]],1]];
rad[n_]:=rad[n]=If[n===0,1,NestWhile[#+1&,rad[n-1]+1,Not[radQ[#]]&]];
Set@@@Array[radPi[rad[#]]==#&,nn];
jfac[n_]:=With[{g=GCD@@FactorInteger[n+1][[All,2]]},JIX[radPi[Power[n+1,1/g]],Flatten[Cases[FactorInteger[g+1],{p_,k_}:>ConstantArray[PrimePi[p],k]]]]];
diwt[n_]:=If[n===1,1,Apply[1+diwt[#1]+Total[diwt/@#2]&,jfac[n-1]]];
Array[diwt,nn]

a(A007916(h)^(A000040(g_1)*...*A000040(g_k)-1)) = 1 + a(h) + a(g_1) + ... + a(g_k).

A114997 Number of ordered trees with n edges and no unary or binary nodes.

Original entry on oeis.org

0, 0, 1, 1, 1, 4, 8, 13, 31, 71, 144, 318, 729, 1611, 3604, 8249, 18803, 42907, 98858, 228474, 528735, 1228800, 2865180, 6693712, 15676941, 36807239, 86584783, 204060509, 481823778, 1139565120, 2699329341, 6403500057, 15211830451, 36183117255, 86171536894, 205459894230, 490417795075

Offset: 1

Nachum Dershowitz, Feb 23 2006

nonn

Also counts sequences of n natural numbers, excluding 1 and 2, such that the sum of every prefix is no more than its length.

a(n) is the number of Dyck paths of semilength n with all ascents of length >= 3. For example, a(6) = 4 counts U^6.D^6, U^3.D.U^3.D^5, U^3.D^2.U^3.D^4, U^3.D^3.U^3.D^3 where ^ denotes repetition and a dot denotes concatenation. - David Callan, Dec 08 2021

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Andrei Asinowski, Axel Bacher, Cyril Banderier, Bernhard Gittenberger, Analytic combinatorics of lattice paths with forbidden patterns, the vectorial kernel method, and generating functions for pushdown automata, Laboratoire d'Informatique de Paris Nord (LIPN 2019).
Nachum Dershowitz and Shmuel Zaks, More patterns in trees: Up and down, young and old, odd and even, SIAM J. Discrete Mathematics, 23 (2009), 447-465.

Cf. A000108 (rev. of x/(1+1*Sum_{k>=1} x^k) ), A005043 (rev. of x/(1+x*Sum_{k>=1} x^k) ), A215341 (rev. of x/(1+x^3*Sum_{k>=1} x^k) ).

eq := x^3*A^3+x*A^2-(1+x)*A+1 = 0: A := RootOf(eq, A): Aser := series(A, x = 0, 40): seq(coeff(Aser, x, n), n = 1 .. 38); # Emeric Deutsch, Jan 13 2015

Table[Sum[1/(n+1)*Binomial[n+1,k]*Binomial[2*k-n-3,n-k],{k,Ceiling[(n+3)/2],n}],{n,1,20}] (* Vaclav Kotesovec, Mar 22 2014 *)

a(n)=sum(k=ceil((n+3)/2), n, (1/(n+1) * binomial(n+1, k) * binomial(2*k-n-3, n-k)) ); \\ Joerg Arndt, Aug 19 2012

N=66; gf=serreverse(x/(1+x^2*sum(k=1,N,x^k))+O(x^N)) / x;
/* = 1 + x^3 + x^4 + x^5 + 4*x^6 + 8*x^7 + 13*x^8 + 31*x^9 + ... */
v114997=Vec(gf) /* = [1, 0, 0, 1, 1, 1, 4, 8, 13, 31, ...] */  \\ Joerg Arndt, Aug 19 2012

a(n) = Sum_{(n+3)/2 <= k <= n} (1/(n+1) * binomial(n+1, k) * binomial(2*k-n-3, n-k)).
If A(x) is the g.f. for the sequence with a(0)=1, then x^3*A^3+x*A^2-(1 + x)*A+1 = 0. - Emeric Deutsch, Jan 13 2015
Let A(x) be the g.f. for the sequence with a(0)=1, then x*A(x) is the reversion of x/(1+x^2*sum(k>=1,x^k)). - Joerg Arndt, Aug 19 2012 (proved by Emeric Deutsch, Jan 13 2015)
Recurrence: (n+1)*(n+2)*(28*n^2 - 38*n - 15)*a(n) = -4*(n+1)*(14*n^3 - 12*n^2 + 7*n - 15)*a(n-1) + (n-2)*(140*n^3 + 90*n^2 - 221*n + 45)*a(n-2) + 6*(n-2)*(28*n^3 - 24*n^2 - 75*n + 95)*a(n-3) + 23*(n-3)*(n-2)*(28*n^2 + 18*n - 25)*a(n-4). - Vaclav Kotesovec, Mar 22 2014
a(n) ~ c / (n^(3/2) * r^n), where r = (4*sqrt(2) - 3 + 23*sqrt((344*sqrt(2))/529 - 235/529))/46 = 0.402505948621022106992... is the root of the equation 23*r^4+6*r^3+5*r^2-2*r-1 = 0 and c = sqrt((280 + 133*sqrt(2) - 25*sqrt(14*(11 + 8*sqrt(2)))) / (7*Pi))/4 = 0.273007516... - Vaclav Kotesovec, Mar 22 2014, updated Jan 14 2015

Offset set to 1 by Joerg Arndt, Aug 19 2012

A358378 Numbers k such that the k-th standard ordered rooted tree is fully canonically ordered (counted by A000081).

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 15, 16, 17, 21, 25, 27, 29, 31, 32, 37, 41, 43, 49, 53, 57, 59, 61, 63, 64, 65, 73, 81, 85, 101, 105, 107, 113, 117, 121, 123, 125, 127, 128, 129, 137, 145, 165, 169, 171, 193, 201, 209, 213, 229, 233, 235, 241, 245, 249, 251

Offset: 1

Gus Wiseman, Nov 14 2022

nonn

The ordering of finitary multisets is first by length and then lexicographically. This is also the ordering used for Mathematica expressions.

We define the n-th standard ordered rooted tree to be obtained by taking the (n-1)-th composition in standard order (graded reverse-lexicographic, A066099) as root and replacing each part with its own standard ordered rooted tree. This ranking is an ordered variation of Matula-Goebel numbers, giving a bijective correspondence between positive integers and unlabeled ordered rooted trees.

			The terms together with their corresponding ordered rooted trees begin:
   1: o
   2: (o)
   3: ((o))
   4: (oo)
   5: (((o)))
   7: (o(o))
   8: (ooo)
   9: ((oo))
  11: ((o)(o))
  13: (o((o)))
  15: (oo(o))
  16: (oooo)
  17: ((((o))))
  21: ((o)((o)))

Gus Wiseman, Statistics, classes, and transformations of standard compositions

These trees are counted by A000081.
A358371 and A358372 count leaves and nodes in standard ordered rooted trees.
Cf. A001263, A004249, A005043, A032027, A063895, A276625, A358373-A358377.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
srt[n_]:=If[n==1,{},srt/@stc[n-1]];
Select[Range[1000],FreeQ[srt[#],[_]?(!OrderedQ[#]&)]&]

A007971 INVERTi transform of central trinomial coefficients (A002426).

Original entry on oeis.org

0, 1, 2, 2, 4, 8, 18, 42, 102, 254, 646, 1670, 4376, 11596, 31022, 83670, 227268, 621144, 1706934, 4713558, 13072764, 36398568, 101704038, 285095118, 801526446, 2259520830, 6385455594, 18086805002, 51339636952, 146015545604

Offset: 0

David Dumas (dumas(AT)TCNJ.EDU)

nonn

Number of paths of a walk on the integers, allowing steps of size 0, +1, and -1, which return to the starting point for the first time at time n. [David P. Sanders (dps(AT)fciencias.unam.mx), May 04 2009]

			G.f. = x + 2*x^2 + 2*x^3 + 4*x^4 + 8*x^5 + 18*x^6 + 42*x^7 + 102*x^8 + 254*x^9 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Alexander Burstein and Louis W. Shapiro, Pseudo-involutions in the Riordan group and Chebyshev polynomials, arXiv:2502.13673 [math.CO], 2025.

Cf. A001006, A002426, A005043.

Cf. A025227.

CoefficientList[Series[1-Sqrt[1-2x-3x^2],{x,0,40}],x] (* Harvey P. Dale, Dec 17 2012 *)
a[1]:=1;a[2]:=2;a[n_]:=a[n]=1/2 Sum[a[k] a[n-k],{k,1,n-1}];
Join[{0},Map[a,Range[24]]] (* Oliver Seipel, Nov 03 2024, after Schröder 1870 *)

x='x+O('x^50); concat([0], Vec(1 - sqrt(1 - 2*x - 3*x^2))) \\ G. C. Greubel, Feb 26 2017

A002426(n) = Sum_{i=1..n} a(i)*A002426(n-i), n>0. - Michael Somos, Jun 14 2000
G.f.: 1 - sqrt(1 - 2*x - 3*x^2). - Michael Somos, Jun 14 2000
a(0)=0, a(1)=1, a(2)=2, then a(n) = (1/2) *(a(1)*a(n-1)+a(2)*a(n-2)+....+a(n-1)*a(1)). - Benoit Cloitre, Oct 24 2003
a(n) = 2^(1-n)*Sum_{k=1..n} (binomial(k,n-k)*A000108(k-1)*3^(n-k)), n>0. - Vladimir Kruchinin, Feb 05 2011
G.f.: 1-sqrt(1-2*x-3*(x^2)) =  x/G(0) ; G(k) = 1-2*x/(1+x/(1+x/(1-2*x/(1-x/(2-x/G(k+1)))))) ; (continued fraction). - Sergei N. Gladkovskii, Dec 11 2011
a(n+2) = 2 * A001006(n). - Michael Somos, Jun 14 2000
For n>1, a(n) = 2 * (A005043(n-1) + A005043(n-2)). - Ralf Stephan, Jul 06 2003
0 = a(n) * (9*a(n+1) + 15*a(n+2) - 12*a(n+3)) + a(n+1) * (-3*a(n+1) + 10*a(n+2) - 5*a(n+3)) + a(n+2) * (a(n+2) + a(n+3)) for all n>0. - Michael Somos, Jan 25 2014
n*a(n) + (-2*n+3)*a(n-1) + *(-n+3)*a(n-2) = 0. - R. J. Mathar, Sep 06 2016

Name corrected by Michael Somos, Mar 23 2012

A039717 Row sums of convolution triangle A030523.

Original entry on oeis.org

1, 4, 15, 55, 200, 725, 2625, 9500, 34375, 124375, 450000, 1628125, 5890625, 21312500, 77109375, 278984375, 1009375000, 3651953125, 13212890625, 47804687500, 172958984375, 625771484375, 2264062500000, 8191455078125

Offset: 1

Wolfdieter Lang

Number of (s(0), s(1), ..., s(2n)) such that 0 < s(i) < 10 and |s(i) - s(i-1)| = 1 for i = 1,2,...,2n, s(0) = 3, s(2n) = 5.
With offset 0 = INVERT transform of A001792: (1, 3, 8, 20, 48, 112, ...). - Gary W. Adamson, Oct 26 2010
From Tom Copeland, Nov 09 2014: (Start)
The array belongs to a family of arrays associated to the Catalan A000108 (t=1), and Riordan, or Motzkin sums A005043 (t=0), with the o.g.f. (1-sqrt(1-4x/(1+(1-t)x)))/2 and inverse x*(1-x)/(1 + (t-1)*x*(1-x)). See A091867 for more info on this family. Here t = -4 (mod signs in the results).
Let C(x) = (1 - sqrt(1-4x))/2, an o.g.f. for the Catalan numbers A000108, with inverse Cinv(x) = x*(1-x) and P(x,t) = x/(1+t*x) with inverse P(x,-t).
O.g.f.: G(x) = x*(1-x)/(1 - 5x*(1-x)) = P(Cinv(x),-5).
Inverse O.g.f.: Ginv(x) = (1 - sqrt(1 - 4*x/(1+5x)))/2 = C(P(x,5)) (signed A026378). Cf. A030528. (End)
p-INVERT of (2^n), where p(s) = 1 - s - s^2; see A289780. - Clark Kimberling, Aug 10 2017

Michel Marcus, Table of n, a(n) for n = 1..1000
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
Index entries for linear recurrences with constant coefficients, signature (5,-5).

Appears in A109106.

Cf. A000045, A000108, A001792, A005043, A091867, A026378, A030528.

CoefficientList[Series[(1 - x) / (1 - 5 x + 5 x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Nov 10 2014 *)

Vec(x*(1-x)/(1-5*x+5*x^2) + O(x^40)) \\ Altug Alkan, Nov 20 2015

G.f.: x*(1-x)/(1-5*x+5*x^2) = g1(3, x)/(1-g1(3, x)), g1(3, x) := x*(1-x)/(1-2*x)^2 (g.f. first column of A030523).
From Paul Barry, Apr 16 2004: (Start)
Binomial transform of Fibonacci(2n+2).
a(n) = (sqrt(5)/2 + 5/2)^n*(3*sqrt(5)/10 + 1/2) - (5/2 - sqrt(5)/2)^n*(3*sqrt(5)/10 - 1/2). (End)
a(n) = (1/5)*Sum_{r=1..9} sin(3*r*Pi/10)*sin(r*Pi/2)*(2*cos(r*Pi/10))^(2*n).
a(n) = 5*a(n-1) - 5*a(n-2).
a(n) = Sum_{k=0..n} Sum_{i=0..n} binomial(n, i)*binomial(k+i+1, 2k+1). - Paul Barry, Jun 22 2004
From Johannes W. Meijer, Jul 01 2010: (Start)
Limit_{k->oo} a(n+k)/a(k) = (A020876(n) + A093131(n)*sqrt(5))/2.
Limit_{n->oo} A020876(n)/A093131(n) = sqrt(5).
(End)
From Benito van der Zander, Nov 19 2015: (Start)
Limit_{k->oo} a(k+1)/a(k) = 1 + phi^2 = (5 + sqrt(5)) / 2.
a(n) = a(n-1) * 3 + A081567(n-2) (not proved).
(End)
E.g.f.: exp(x*5/2) * (cosh(x*sqrt(5)/2) + (3/sqrt(5))*sinh(x*sqrt(5)/2)). - Fabian Pereyra, Oct 29 2024

A100754 Triangle read by rows: T(n, k) = number of hill-free Dyck paths (i.e., no peaks at height 1) of semilength n and having k peaks.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 8, 8, 1, 1, 13, 29, 13, 1, 1, 19, 73, 73, 19, 1, 1, 26, 151, 266, 151, 26, 1, 1, 34, 276, 749, 749, 276, 34, 1, 1, 43, 463, 1781, 2762, 1781, 463, 43, 1, 1, 53, 729, 3758, 8321, 8321, 3758, 729, 53, 1, 1, 64, 1093, 7253, 21659, 31004, 21659, 7253, 1093, 64, 1

Offset: 2

Emeric Deutsch, Jan 14 2005

Row n has n - 1 terms. Row sums yield the Fine numbers (A000957).
Related to the number of certain sets of non-crossing partitions for the root system A_n (p. 11, Athanasiadis and Savvidou). - Tom Copeland, Oct 19 2014
T(n,k) is the number of permutations pi of [n-1] with k - 1 descents such that s(pi) avoids the patterns 132, 231, and 312, where s is West's stack-sorting map. - Colin Defant, Sep 16 2018
The absolute values of the polynomials at -1 and j (cube root of 1) seem to be given by A126120 and A005043. - F. Chapoton, Nov 16 2021
Don Knuth observes that this sequence also arrises from the enumeration of restricted max-and-min-closed relations, only there it appears as an array read by antidiagonals: see the Knuth "Notes" link and A372068. Knuth also gives a formula expressing the array A372368 in terms of this array. He also reports that there is strong experimental evidence that the n-th term of row m in this array is a polynomial of degree 2*m-2 in n. - N. J. A. Sloane, May 12 2024

			T(4, 2) = 4 because we have UU*DDUU*DD, UU*DUU*DDD, UUU*DDU*DD and UUU*DU*DDD, where U = (1, 1), D = (1,-1) and * indicates the peaks.
Triangle starts:
   1;
   1,  1;
   1,  4,   1;
   1,  8,   8,    1;
   1, 13,  29,   13,    1;
   1, 19,  73,   73,   19,    1;
   1, 26, 151,  266,  151,   26,    1;
   1, 34, 276,  749,  749,  276,   34,   1;
   1, 43, 463, 1781, 2762, 1781,  463,  43,  1;
   1, 53, 729, 3758, 8321, 8321, 3758, 729, 53, 1;
   ...
As an array (for which the rows of the preceding triangle are the antidiagonals):
   1,  1,    1,     1,      1,      1,       1,        1,        1, ...
   1,  4,    8,    13,     19,     26,      34,       43,       53, ...
   1,  8,   29,    73,    151,    276,     463,      729,     1093, ...
   1, 13,   73,   266,    749,   1781,    3758,     7253,    13061, ...
   1, 19,  151,   749,   2762,   8321,   21659,    50471,   107833, ...
   1, 26,  276,  1781,   8321,  31004,   97754,   271125,   679355, ...
   1, 34,  463,  3758,  21659,  97754,  367285,  1196665,  3478915, ...
   1, 43,  729,  7253,  50471, 271125, 1196665,  4526470, 15118415, ...
   1, 53, 1093, 13061, 107833, 679355, 3478915, 15118415, 57500480, ...
   ...

Alois P. Heinz, Rows n = 2..142, flattened
C. Athanasiadis and C. Savvidou, The local h-vector of the cluster subdivision of a simplex, arXiv preprint arXiv:1204.0362 [math.CO], 2012.
Paul Barry, Three Études on a sequence transformation pipeline, arXiv:1803.06408 [math.CO], 2018.
Colin Defant, Stack-sorting preimages of permutation classes, arXiv:1809.03123 [math.CO], 2018.
E. Deutsch and L. Shapiro, A survey of the Fine numbers, Discrete Math., 241 (2001), 241-265.
D. E. Knuth, Notes on four arrays of numbers arising from the enumeration of CRC constraints and min-and-max-closed constraints, May 06 2024

Cf. A000957, A108263, A132081.

See also A099594, A272644, A372066, A372067, A372068.

T := (n, k) -> add((j/(n-j))*binomial(n-j, k-j)*binomial(n-j,k), j=0..min(k,n-k)): for n from 2 to 13 do seq(T(n, k), k = 1..n-1) od; # yields the sequence in triangular form

T[n_, k_] := Sum[(j/(n-j))*Binomial[n-j, k-j]*Binomial[n-j, k], {j, 0, Min[k, n-k]}]; Table[T[n, k], {n, 2, 13}, {k, 1, n-1}] // Flatten (* Jean-François Alcover, Feb 19 2017, translated from Maple *)

T(n, k) = Sum_{j=0..min(k, n-k)} (j/(n-j)) * C(n-j, k-j) * C(n-j, k), n >= 2.
G.f.: t*z*r/(1 - t*z*r), where r = r(t, z) is the Narayana function defined by r = z*(1 + r)*(1 + t*r).
From Tom Copeland, Oct 19 2014: (Start)
With offset 0 for A108263 and offset 1 for A132081, row polynomials of this entry P(n, x) = Sum_{i} A108263(n, i)*x^i*(1 + x)^(n - 2*i) = Sum_{i} A132081(n - 2, i)*x^i*(1 + x)^(n - 2*i).
E.g., P(4, x) = 1*x*(1 + x)^(4 - 2*1) + 2*x^2*(1 + x)^(4 - 2*2) = x + 4*x^2 + x^3.
Equivalently, let Q(n, x) be the row polynomials of A108263. Then P(n, x) = (1 + x)^n * Q(n, x/(1 + x)^2).
E.g., P(4, x) = (1 + x)^4 * (x/(1 + x)^2 + 2 * (x/(1 + x)^2)^2).
See Athanasiadis and Savvidou (p. 7). (End)

A104597 Triangle T read by rows: inverse of Motzkin triangle A097609.

Original entry on oeis.org

1, 0, 1, -1, 0, 1, -1, -2, 0, 1, 0, -2, -3, 0, 1, 1, 1, -3, -4, 0, 1, 1, 4, 3, -4, -5, 0, 1, 0, 3, 9, 6, -5, -6, 0, 1, -1, -2, 5, 16, 10, -6, -7, 0, 1, -1, -6, -9, 6, 25, 15, -7, -8, 0, 1, 0, -4, -18, -24, 5, 36, 21, -8, -9, 0, 1, 1, 3, -7, -39, -50, 1, 49, 28, -9, -10, 0, 1, 1, 8

Offset: 0

Ralf Stephan, Mar 17 2005

Riordan array ((1-x)/(1-x+x^2),x(1-x)/(1-x+x^2)). - Paul Barry, Jun 21 2008

			1
0,1
-1,0,1
-1,-2,0,1
0,-2,-3,0,1
1,1,-3,-4,0,1
1,4,3,-4,-5,0,1
0,3,9,6,-5,-6,0,1
-1,-2,5,16,10,-6,-7,0,1
-1,-6,-9,6,25,15,-7,-8,0,1

D. Merlini, R. Sprugnoli and M. C. Verri, An algebra for proper generating trees, Mathematics and Computer Science, Part of the series Trends in Mathematics pp 127-139, 2000. [alternative link]
D. Merlini, R. Sprugnoli and M. C. Verri, An algebra for proper generating trees, Colloquium on Mathematics and Computer Science, Versailles, September 2000.

Row sums are A009116 with different signs.
Row sums are A146559(n).
Cf. A005043, A091867, A030528, A000108.

# Uses function InvPMatrix from A357585. Adds column 1, 0, 0, ... to the left.
InvPMatrix(10, n -> A005043(n-1)); # Peter Luschny, Oct 09 2022

T(n,m):=sum(binomial(m,j)*sum(binomial(k,n-k)*(-1)^(n-k)*binomial(k+j-1,j-1),k,0,n)*(-1)^(m-j),j,0,m); /* Vladimir Kruchinin, Apr 08 2011 */

T(n,m) = sum(j=0..m, binomial(m,j)*sum(k=0..n, binomial(k,n-k)*(-1)^(n-k)*binomial(k+j-1,j-1))*(-1)^(m-j)). - Vladimir Kruchinin, Apr 08 2011
T(n,m) = sum(k=ceiling((n-m-1)/2)..n-m, binomial(k+m,m)*binomial(k+1,n-k-m)*(-1)^(n-k-m)). - Vladimir Kruchinin, Dec 17 2011
T(n,k) = T(n-1,k) + T(n-1,k-1) - T(n-2,k) - T(n-2,k-1), T(0,0) = T(1,1) = 1, T(1,0) = 0, T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Feb 20 2013
T(n+5,n) = (n+1)^2. - Philippe Deléham, Feb 20 2013
From Tom Copeland, Nov 04 2014: (Start)
O.g.f.: G(x,t) = Pinv[Cinv(x),t+1] = Cinv(x) / [1 - (t+1)Cinv(x)] = x*(1-x) / [1-(t+1)x(1-x)] = x + t * x^2 + (-1 + t^2) * x^3 + ..., where Cinv(x)= x * (1-x) is the inverse of C(x) = [1-sqrt(1-4*x)]/2, an o.g.f. for the Catalan numbers A000108 and Pinv(x,t) = -P(-x,t) = x/(1-t*x) is the inverse of P(x,t) = x/(1+x*t).
Ginv(x,t)= C[P[x,t+1]]= C[x/(1+(t+1)x)] = {1-sqrt[1-4*x/(1+(t+1)x)]}/2.
The inverse in x of G(x,t) with t replaced by -t is the o.g.f. of A091867, and G(x,t-1) is a signed version of the (mirrored) Fibonacci polynomials A030528. (End)

cp's OEIS Frontend

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A279944 Number of positions in the free pure symmetric multifunction in one symbol with j-number n.

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A099323 Expansion of (sqrt(1+3x) + sqrt(1-x))/(2sqrt(1-x)).