A122888 -id:A122888 - cp's OEIS frontend

A152798 Triangle defined by T(n,k) = Sum_{j=0..k} C(k,j)*T(n-1,j+k) for n>k>0 with T(n,0)=T(n,n)=1, read by rows.

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 3, 1, 1, 1, 6, 6, 4, 1, 1, 1, 12, 15, 10, 5, 1, 1, 1, 27, 40, 29, 15, 6, 1, 1, 1, 67, 113, 93, 49, 21, 7, 1, 1, 1, 180, 348, 310, 180, 76, 28, 8, 1, 1, 1, 528, 1148, 1106, 685, 311, 111, 36, 9, 1, 1, 1, 1676, 4045, 4205, 2748, 1322, 497, 155, 45

Offset: 0

Paul D. Hanna, Dec 23 2008

			Triangle begins:
1;
1, 1;
1, 1, 1;
1, 2, 1, 1;
1, 3, 3, 1, 1;
1, 6, 6, 4, 1, 1;
1, 12, 15, 10, 5, 1, 1;
1, 27, 40, 29, 15, 6, 1, 1;
1, 67, 113, 93, 49, 21, 7, 1, 1;
1, 180, 348, 310, 180, 76, 28, 8, 1, 1;
1, 528, 1148, 1106, 685, 311, 111, 36, 9, 1, 1;
1, 1676, 4045, 4205, 2748, 1322, 497, 155, 45, 10, 1, 1;
1, 5721, 15203, 16912, 11683, 5858, 2323, 750, 209, 55, 11, 1, 1;
1, 20924, 60710, 71858, 52262, 27349, 11230, 3809, 1083, 274, 66, 12, 1, 1; ...
ILLUSTRATE RECURRENCE:
T(6,1) = T(5,1) + T(5,2) = 6 + 6 = 12;
T(7,2) = T(6,2) + 2*T(6,3) + T(6,4) = 6 + 2*4 + 1 = 15;
T(8,3) = T(7,3) + 3*T(7,4) + 3*T(7,5) + T(7,6) = 29 + 3*15 + 3*6 + 1 = 93.
Note that column 1 equals A122889: [1,1,2,3,6,12,27,67,180,528,...]
which is the antidiagonal sums of triangle A122888.
RELATED TRIANGLE A122888 begins:
1;
1, 1;
1, 2, 2, 1;
1, 3, 6, 9, 10, 8, 4, 1;
1, 4, 12, 30, 64, 118, 188, 258, 302, 298, 244, 162, 84, 32, 8, 1;
1, 5, 20, 70, 220, 630, 1656, 4014, 8994, 18654, 35832, 63750,...;
1, 6, 30, 135, 560, 2170, 7916, 27326, 89582, 279622, 832680,...;
1, 7, 42, 231, 1190, 5810, 27076, 121023, 520626, 2161158,...;
1, 8, 56, 364, 2240, 13188, 74760, 409836, 2179556, 11271436,...; ...
in which the g.f. of row n equals the n-th iteration of (x+x^2).

Cf. A122888; columns: A122889, A152799; variant: A101494.

PARI
```
T(n, k)=if(n
				
```

A122938 G.f. A(x) satisfies: A(x+x^2) = A(x)^2/(1+x).

Original entry on oeis.org

1, 1, 1, 2, 6, 27, 160, 1189, 10600, 110161, 1306629, 17408293, 257299241, 4177017722, 73872560359, 1413560616317, 29096001945172, 641010535303531, 15049350893772391, 375084409475304164, 9890697492431533299

Offset: 0

Paul D. Hanna, Sep 21 2006

nonn

Self-convolution equals A122939. See A122888 for the table of self-compositions of x+x^2.

			G.f.: A(x) = (1 + x)^(1/2) * (1 + x+x^2)^(1/4) * (1 + x+2x^2+2x^3+x^4)^(1/8) * (1 + x+3x^2+6x^3+9x^4+10x^5+8x^6+4x^7+x^8)^(1/16) *...

Cf. A122939 (A^2), A122940 (log), A122941-A122945; A122888 (table).

{a(n)=local(A=1+x+x*O(x^n)); for(i=0,n,A=-A+2*sqrt((1+x)*subst(A,x,x+x^2+x*O(x^n))));polcoeff(A,n)}

G.f.: A(x) = Product_{n>=0} (1 + F_n(x) )^(1/2^(n+1)) where F_0(x)=x, F_{n+1}(x)=F_n(x+x^2); a product that involves the n-th self-compositions of x+x^2.

A122940 L.g.f.: A(x) satisfies: A(x+x^2) = 2*A(x) - log(1+x) with A(0)=0; thus A(x) = log(B(x)), where B(x) is g.f. of A122938.

Original entry on oeis.org

1, 1, 4, 17, 106, 796, 7176, 75057, 894100, 11946906, 176939192, 2876683340, 50931297912, 975391344376, 20090039762944, 442830738561585, 10400937450758286, 259318357362882148, 6839990934297006668

Offset: 1

Paul D. Hanna, Sep 25 2006

nonn

a(n) = n * Sum_{k=0..n-1} (-1)^(n-k-1)*A122941(n-k,k)/(n-k).

			To illustrate A(x+x^2) = 2*A(x) - log(1+x):
A(x) = x + 1*x^2/2 + 4*x^3/3 + 17*x^4/4 + 106*x^5/5 + 796*x^6/6 +...
A(x+x^2) = x + 3*x^2/2 + 7*x^3/3 + 35*x^4/4 + 211*x^5/5 + 1593*x^6/6 +...

Cf. A122938; related tables: A122941, A122888.

{a(n)=local(A=x+x*O(x^n)); for(i=0,n,A=-A+subst(A,x,x+x^2)+log(1+x+x*O(x^n)));n*polcoeff(A,n)}

L.g.f.: A(x) = Sum_{n>=1} a(n)*x^n/n = Sum_{n>=0} log(1 + F_n(x))/2^(n+1) where F_0(x)=x, F_{n+1}(x) = F_n(x+x^2); a sum involving self-compositions of x+x^2 (cf. A122888).

A122941 Rectangular table, read by antidiagonals, where the g.f. of row n is Sum_{i>=0} F_i(x)^n / 2^(i+1), where F_0(x)=x, F_{n+1}(x) = F_n(x+x^2), for n>=1.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 7, 7, 1, 4, 15, 34, 34, 1, 5, 26, 94, 214, 214, 1, 6, 40, 200, 726, 1652, 1652, 1, 7, 57, 365, 1831, 6645, 15121, 15121, 1, 8, 77, 602, 3865, 19388, 70361, 160110, 160110, 1, 9, 100, 924, 7239, 46481, 233154, 846144, 1925442, 1925442, 1

Offset: 1

Paul D. Hanna, Sep 25 2006

A122940(n)/n = Sum_{m=1..n} (-1)^(m-1)*T(m,n-m+1)/m ; where l.g.f. of A122940, L(x), satisfies: L(x+x^2) = 2*L(x) - log(1+x).

			Table begins:
1, 1, 2, 7, 34, 214, 1652, 15121, 160110, 1925442, 25924260, ...;
1, 2, 7, 34, 214, 1652, 15121, 160110, 1925442, 25924260, ...;
1, 3, 15, 94, 726, 6645, 70361, 846144, 11392530, 169785124, ...;
1, 4, 26, 200, 1831, 19388, 233154, 3139200, 46784118, ...;
1, 5, 40, 365, 3865, 46481, 625820, 9326720, 152426170, ...;
1, 6, 57, 602, 7239, 97470, 1452610, 23739936, 422171622, ...;
1, 7, 77, 924, 12439, 185388, 3029782, 53879148, 1035760670, ...;
1, 8, 100, 1344, 20026, 327296, 5820360, 111889248, 2312153223, ...;
1, 9, 126, 1875, 30636, 544824, 10473576, 216432783, 4784414985, ...;
1, 10, 155, 2530, 44980, 864712, 17868995, 395007850, 9301284465, ...;
Given that A122940 begins:
[1, 1, 4, 17, 106, 796, 7176, 75057, 894100, 11946906, ...],
demonstrate A122940(n)/n = Sum_{m=1..n} (-1)^(m-1)*T(m,n-m+1)/m
at n=4: A122940(4)/4 = 17/4 = 7/1 - 7/2 + 3/3 - 1/4;
at n=5: A122940(5)/5 = 106/5 = 34/1 - 34/2 + 15/3 - 4/4 + 1/5;
at n=6: A122940(6)/6 = 796/6 = 214/1 - 214/2 + 94/3 - 26/4 + 5/5 - 1/6.

Cf. A122940; rows: A122942, A122943, A122944, A122945; related tables: A122888, A122946, A122948, A122951.

/* Get T(n,k) from H(n,), the n-th self-composition of x+x^2: */
{H(n,p)=local(F=x+x^2, G=x+x*O(x^p));if(n==0,G=x,for(i=1,n,G=subst(F,x,G));G)}
{T(n,k)=round(polcoeff( sum(i=0,6*n+100,H(i,k+n-1)^n/2^(i+1)),k+n-1))}

T(n,k) = [x^k] Sum_{i>=0} F_i(x)^n / 2^(i+1) where F_0(x)=x, F_{n+1}(x) = F_n(x+x^2); a sum involving n-th powers of self-compositions of x+x^2 (cf. A122888).

A122891 Column sums of triangle A122890.

Original entry on oeis.org

1, 1, 3, 28, 1625, 3247268, 10649997137454, 113423713055030979289411081, 12864938683278671740537137672878980378983810317967737

Offset: 0

Paul D. Hanna, Sep 18 2006

nonn

Cf. A122890, A122888; A007018.

{A122890(n, k)=local(F=x+x^2, G=x+x*O(x^k)); if(n<0, 0, for(i=1, n, G=subst(F, x, G)); polcoeff(G, k, x))}
/* Takes first 2^m terms in column m of A122890 to compute a(m): */
{for(m=0,8,print1(polcoeff(sum(N=1,2^m,sum(n=0,2^m,A122890(n,N)*x^n)*(1-x)^N+O(x^12)),m),","))}

A122893 Position of largest coefficient of n-th self-composition of (x+x^2) for n>=0.

Original entry on oeis.org

1, 1, 2, 5, 9, 19, 37, 74, 147, 294, 587, 1175, 2349, 4698, 9396, 18791, 37582

Offset: 0

Paul D. Hanna, Sep 19 2006

What is the limit a(n)/2^n = 0.573... ? Originated by Ralf Stephan in A092123 as the position of largest coefficient in the expansion of P(0)=x, P(n+1)=P(n)*[1+P(n)] (equivalent definition).

Cf. A092123, A122894; A122888.

P[n_] := P[n] = If[n < 1, x, P[n - 1]*(P[n - 1] + 1)]; Table[p = Expand[CoefficientList[P[n], x]]; Position[p, Max[p]][[1]][[1]] - 1, {n, 0, 12}] (* Vaclav Kotesovec, Nov 08 2018 *)

a(14)-a(15) from Vaclav Kotesovec, Nov 08 2018

a(16) from Vaclav Kotesovec, Nov 09 2018

A122894 Coefficient of x^(2^(n-1)) in the n-th self-composition of (x+x^2) for n>=1.

Original entry on oeis.org

1, 2, 9, 258, 293685, 531124770570, 2439717292075827330588969, 72554628124279239546273779187960042205300343234178

Offset: 1

Paul D. Hanna, Sep 19 2006

nonn

Originated by Ralf Stephan in A092123 as the 2^(n-1)th coefficient in the expansion of P(0)=x, P(n+1)=P(n)*[1+P(n)] (equivalent definition). Next term is too large to include.

			a(1) = 1 = [x^1] (x + x^2).
a(2) = 2 = [x^2] (x + 2*x^2 + 2*x^3 + x^4).
a(3) = 9 = [x^4] (x + 3*x^2 + 6*x^3 + 9*x^4 + 10*x^5 + 8*x^6 + 4*x^7 + x^8).

Cf. A092123, A122893; A122888.

{a(n)=local(F=x+x^2, G=x+x*O(x^(2^(n-1)))); if(n<1, 0, for(i=1, n, G=subst(F, x, G)); return(polcoeff(G, 2^(n-1), x)))}

A122939 G.f. A(x) satisfies: A(x+x^2) = A(x)^2/(1+x)^2.

Original entry on oeis.org

1, 2, 3, 6, 17, 70, 390, 2776, 24042, 244864, 2862185, 37715474, 552685976, 8910951840, 156709821779, 2984589501562, 61188398397436, 1343410717573876, 31445844702847347, 781689483100388326, 20564696601659697997

Offset: 0

Paul D. Hanna, Sep 21 2006

nonn

Self-convolution of A122938. See A122888 for the table of self-compositions of x+x^2.

			G.f.: A(x) = (1 + x) * (1 + x+x^2)^(1/2) * (1 + x+2x^2+2x^3+x^4)^(1/4) * (1 + x+3x^2+6x^3+9x^4+10x^5+8x^6+4x^7+x^8)^(1/8) *...

Cf. A122938 (square-root), A122888 (table).

{a(n)=local(A=1+x+x*O(x^n)); for(i=0,n,A=-A+2*(1+x)*sqrt(subst(A,x,x+x^2+x*O(x^n))));polcoeff(A,n)}

G.f.: A(x) = Product_{n>=0} (1 + F_n(x) )^(1/2^n) where F_0(x)=x, F_{n+1}(x)=F_n(x+x^2); a product that involves the n-th self-compositions of x+x^2.

A166904 Row sums of triangle A166900.

Original entry on oeis.org

1, 2, 7, 40, 321, 3361, 43667, 679806, 12358885, 257281501, 6039232167, 157879127902, 4550258562799, 143367509714352, 4903128661348411, 180907738215049666, 7163333648262397913, 303006716530386750233

Offset: 0

Paul D. Hanna, Nov 27 2009

nonn

Triangle A166900 transforms rows into diagonals in the table of coefficients of successive iterations of x+x^2 (cf. A122888).

Cf. A166900, A166901, A166902, A166903, A122888, A135080.

{a(n)=local(F=x, M, N, P); M=matrix(n+2, n+2, r, c, F=x;for(i=1, r+c-2, F=subst(F, x, x+x^2+x*O(x^(n+2)))); polcoeff(F, c)); N=matrix(n+1, n+1, r, c, F=x;for(i=1, r, F=subst(F, x, x+x^2+x*O(x^(n+3)))); polcoeff(F, c)); P=matrix(n+1, n+1, r, c, M[r+1, c]); M=(P~*N~^-1); sum(k=1,n+1,M[n+1,k])}

A171792 G.f. A(x) satisfies: A(x) = (x + A(x+x^2))/2 with A(0)=0.

Original entry on oeis.org

1, 1, 2, 7, 34, 214, 1652, 15121, 160110, 1925442, 25924260, 386354366, 6314171932, 112286067892, 2158562109096, 44605949528355, 986049177712850, 23218586050641090, 580198948211652348, 15334750335623526670, 427408226085246086676, 12528910074528593086980

Offset: 1

Paul D. Hanna, Jan 25 2010

nonn

			G.f.: A(x) = x + x^2 + 2*x^3 + 7*x^4 + 34*x^5 + 214*x^6 +...
A(x+x^2) = x + 2*x^2 + 4*x^3 + 14*x^4 + 68*x^5 + 428*x^6 + ...

Alois P. Heinz, Table of n, a(n) for n = 1..425 (first 140 terms from Vaclav Kotesovec)
Magnus Aspenberg, Rodrigo Perez, Control of cancellations that restrain the growth of a binomial recursion, arXiv:1006.1340 [math.CO], 2010; DOI:10.1007/s12220-014-9489-y, The Journal of Geometric Analysis, Vol. 25, No. 3 (2015), 1666-1700.
Olivier Bodini, Antoine Genitrini, Bernhard Gittenberger, On the number of increasing trees with label repetitions, arXiv:1809.04314 [math.CO], 2018.
Olivier Bodini, Antoine Genitrini, Cécile Mailler, Mehdi Naima, Strict monotonic trees arising from evolutionary processes: combinatorial and probabilistic study, hal-02865198 [math.CO] / [math.PR] / [cs.DS] / [cs.DM], 2020.

Cf. A122888.

Nest[Append[#1, Sum[Binomial[k, #2 - k] #[[k]], {k, Floor[#2/2], #2 - 1}]] & @@ {#, Length@ # + 1} &, {1}, 19] (* Michael De Vlieger, Dec 06 2018 *)

a(n):=if n=1 then 1 else sum(binomial(k,n-k)*a(k),k,floor(n/2),n-1); /* Vladimir Kruchinin, Jun 25 2011 */

{a(n)=local(A=x+x^2);for(i=1,n*(n+1)/2,A=(x+subst(A,x,x+x^2+x*O(x^n)))/2);ceil(polcoeff(A,n))}

{a(n)=if(n==1,1,polcoeff(sum(m=1,n-1,a(m)*(x+x^2+x*O(x^n))^m),n))} \\ Paul D. Hanna, Jan 30 2010

G.f.: A(x) = Sum_{n>=0} G_{n}(x)/2^(n+1) where G_{n}(x) is the n-th iteration of (x+x^2) defined by G_{n}(x) = G_{n-1}(x+x^2) with G_0(x)=x.
a(k) = Sum_{n>=0} A122888(n,k)/2^(n+1).
a(k) is odd iff k is a power of 2: a(2^n) == 1 (mod 2) for n>=0.
Conjecture: a(n) = Sum_{r=ceiling(n/2)..n-1} binomial(r, n-r)*a(r) with a(1) = 1. See [Aspenberg, Perez]. - Michel Marcus, Jun 26 2019

cp's OEIS Frontend

A152798 Triangle defined by T(n,k) = Sum_{j=0..k} C(k,j)*T(n-1,j+k) for n>k>0 with T(n,0)=T(n,n)=1, read by rows.

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A122938 G.f. A(x) satisfies: A(x+x^2) = A(x)^2/(1+x).

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A122940 L.g.f.: A(x) satisfies: A(x+x^2) = 2*A(x) - log(1+x) with A(0)=0; thus A(x) = log(B(x)), where B(x) is g.f. of A122938.

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A122941 Rectangular table, read by antidiagonals, where the g.f. of row n is Sum_{i>=0} F_i(x)^n / 2^(i+1), where F_0(x)=x, F_{n+1}(x) = F_n(x+x^2), for n>=1.

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A122891 Column sums of triangle A122890.

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A122893 Position of largest coefficient of n-th self-composition of (x+x^2) for n>=0.

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Mathematica

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A122894 Coefficient of x^(2^(n-1)) in the n-th self-composition of (x+x^2) for n>=1.

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A122939 G.f. A(x) satisfies: A(x+x^2) = A(x)^2/(1+x)^2.

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A166904 Row sums of triangle A166900.

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