A112317 -id:A112317 - cp's OEIS frontend

A122888 Triangle, read by rows, where row n lists the coefficients of x^k, k=1..2^n, in the n-th iteration of (x + x^2) for n>=0.

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, 105024, 160120, 225696, 293685, 352074, 387820, 391232, 359992, 300664, 226580

Offset: 0

Paul D. Hanna, Sep 18 2006

T(n, k) is the number of strings of length k-1 on the alphabet {1, 2, ..., n} such that between every two occurrences of a letter i there is an occurrence of a letter strictly larger than i. For example, for n = 3, k = 4 we have the strings 121, 131, 232 and the six permutations of 123. - Joel B. Lewis, May 06 2008

			Triangle 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,...;
  1, 9, 72, 540, 3864, 26628, 177744, 1153740, 7303164, 45179508,...;
  1, 10, 90, 765, 6240, 49260, 378312, 2836548, 20817588,...; ...
Multiplying the g.f. of column k by (1-x)^k, k>=1, with leading zeros,
 yields the g.f. of row k in the triangle A122890:
  1;
  0, 1;
  0, 0, 2;
  0, 0, 1, 5;
  0, 0, 0, 10, 14;
  0, 0, 0, 8, 70, 42;
  0, 0, 0, 4, 160, 424, 132;
  0, 0, 0, 1, 250, 1978, 2382, 429;
  0, 0, 0, 0, 302, 6276, 19508, 12804, 1430; ...
in which the main diagonal is the Catalan numbers,
 and the row sums form the factorials.

Paul D. Hanna, Table of n, a(n) for n = 0..2046 (rows 0..10)
Art of Problem Solving forum, Strings on [n] with certain restrictions.
Eunmi Choi, Obtuse matrix of arithmetic table, East Asian Math. J. (2024) Vol. 40, No. 3, 329-339. See p. 11.

Cf. A007018 (row sums), diagonals: A112317, A112319, A122887; A092123 (largest term in row); A122889 (antidiagonal sums); A122890 (related triangle).

b:= proc(n) option remember; `if`(n=0, x,
      expand((x-> x+x^2)(b(n-1))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(b(n)):
seq(T(n), n=0..5);  # Alois P. Heinz, Mar 14 2016

f[0][x_] = x; f[n_][x_] := f[n][x] = f[n-1][x+x^2]; row[n_] := CoefficientList[f[n][x], x] // Rest; Table[row[n], {n, 0, 5} ] // Flatten (* Jean-François Alcover, Sep 10 2012 *)

T(m,n):=if m=0 and n=1 then 1 else if m=0 and n>1 then 0 else  if m=1 then binomial(1,n-1) else sum(binomial(i,n-i)*T(m-1,i),i,1,n); /* Vladimir Kruchinin, May 19 2012 */

{T(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)); return(polcoeff(G, k, x)))}
for(n=0, 6, for(k=1, 2^n, print1(T(n, k), ", ")); print(""))

T(n,k) = [x^k] F_n(x) where F_{n+1}(x) = F_n(x+x^2) for n>=1, with F_0(x)=x.

Name changed slightly by Paul D. Hanna, Apr 29 2013

A135080 Triangle, read by rows, that transforms diagonals in the table of coefficients in the successive iterations of x+x^2 (cf. A122888).

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 8, 7, 3, 1, 50, 40, 15, 4, 1, 436, 326, 112, 26, 5, 1, 4912, 3492, 1128, 240, 40, 6, 1, 68098, 46558, 14373, 2881, 440, 57, 7, 1, 1122952, 744320, 221952, 42604, 6135, 728, 77, 8, 1, 21488640, 13889080, 4029915, 748548, 103326, 11565, 1120, 100

Offset: 0

Paul D. Hanna, Nov 18 2007

			Triangle begins:
1;
1, 1;
2, 2, 1;
8, 7, 3, 1;
50, 40, 15, 4, 1;
436, 326, 112, 26, 5, 1;
4912, 3492, 1128, 240, 40, 6, 1;
68098, 46558, 14373, 2881, 440, 57, 7, 1;
1122952, 744320, 221952, 42604, 6135, 728, 77, 8, 1;
21488640, 13889080, 4029915, 748548, 103326, 11565, 1120, 100, 9, 1; ...
Coefficients in iterations of (x+x^2) form table A122888:
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,...; ...
This triangle T transforms one diagonal in the above table into another;
start with the main diagonal of A122888, A112319, which begins:
[1, 1, 2, 9, 64, 630, 7916, 121023, 2179556, 45179508, ...];
then the transform T*A112319 equals A112317, which begins:
[1, 2, 6, 30, 220, 2170, 27076, 409836, 7303164, 149837028, ...];
and the transform T*A112317 equals A112320, which begins:
[1, 3, 12, 70, 560, 5810, 74760, 1153740, 20817588, 430604724, ...].

Paul D. Hanna, Table of n, a(n) for n=0..495 (rows 0..30)

Cf. columns: A135081, A135082, A135083.
Cf. related tables: A122888, A166900, A187005, A187115, A187120.
Cf. related sequences: A112319, A112317, A112320, A187009.

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

/* Generate by method given in A187005, A187115, A187120 (faster): */
{T(n,k)=local(Ck=x);for(m=1,n-k+1,Ck=(1/x^k)*subst(truncate(x^k*Ck),x,x+x^2 +x*O(x^m)));polcoeff(Ck,n-k+1,x)}

Columns may be generated by a method illustrated by triangles A187005, A187115, and A187120. The main diagonal of triangles A187005, A187115, and A187120, equals columns 0, 1, and 2, respectively.

Added cross-reference; example corrected and name changed by Paul D. Hanna, Feb 04 2011

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

Original entry on oeis.org

1, 1, 1, 2, 4, 1, 9, 21, 9, 1, 64, 156, 84, 16, 1, 630, 1540, 935, 230, 25, 1, 7916, 19160, 12480, 3564, 510, 36, 1, 121023, 288813, 196623, 61845, 10465, 987, 49, 1, 2179556, 5123608, 3591560, 1207696, 228800, 25864, 1736, 64, 1, 45179508, 104657520

Offset: 0

Paul D. Hanna, Nov 27 2009

Compare to the triangle A071207 that transforms rows into diagonals in the table of iterations of x/(1-x), where A071207(n,k) gives the number of labeled free trees with n vertices and k children of the root that have a label smaller than the label of the root. Does this triangle have a similar interpretation?

			Triangle begins:
1;
1, 1;
2, 4, 1;
9, 21, 9, 1;
64, 156, 84, 16, 1;
630, 1540, 935, 230, 25, 1;
7916, 19160, 12480, 3564, 510, 36, 1;
121023, 288813, 196623, 61845, 10465, 987, 49, 1;
2179556, 5123608, 3591560, 1207696, 228800, 25864, 1736, 64, 1;
45179508, 104657520, 74847168, 26415840, 5426949, 695079, 56511, 2844, 81, 1;
1059312264, 2420186616, 1755406674, 642448632, 140247810, 19683060, 1830080, 112520, 4410, 100, 1; ...
Coefficients in self-compositions of (x+x^2) form table A122888:
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,...; ...
This triangle T transforms rows of A122888 into diagonals of A122888;
the initial diagonals begin:
A112319: [1, 1, 2, 9, 64, 630, 7916, 121023, 2179556, 45179508, ...];
A112317: [1, 2, 6, 30, 220, 2170, 27076, 409836, 7303164, 149837028,..];
A112320: [1, 3, 12, 70, 560, 5810, 74760, 1153740, 20817588, 430604724, ...].
For example:
T * [1, 0, 0, 0, 0, 0, 0,...]~ = A112319;
T * [1, 1, 0, 0, 0, 0, 0,...]~ = A112317;
T * [1, 2, 2, 1, 0, 0, 0,...]~ = A112320.

Paul D. Hanna, Table of n, a(n) for n=0..495 (rows 0..30)

Cf. A112319, A166901, A166902, A166903, A122888, A135080.

{T(n, k)=local(F=x, M, N, P, m=n); M=matrix(m+2, m+2, r, c, F=x;for(i=1, r+c-2, F=subst(F, x, x+x^2+x*O(x^(m+2)))); polcoeff(F, c)); N=matrix(m+1, m+1, r, c, F=x;for(i=1, r, F=subst(F, x, x+x^2+x*O(x^(m+2)))); polcoeff(F, c)); P=matrix(m+1, m+1, r, c, M[r+1, c]); (P~*N~^-1)[n+1, k+1]}
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))

A119820 Coefficients of x^n in the n-th iteration of x*(1+x)^2 for n>=1.

Original entry on oeis.org

1, 4, 27, 300, 4790, 101010, 2660028, 84191772, 3115739358, 132074618544, 6311492388432, 335744715016854, 19678501474466211, 1260060524755139120, 87519840721085385096, 6553840567691077634748, 526360263009035464610574

Offset: 1

Paul D. Hanna, Jun 01 2006

nonn

			The successive iterations of F(x) = x*(1+x)^2 begin:
F(x) = (1)x + 2x^2 + x^3
F(F(x)) = x + (4)x^2 + 10x^3 + 18x^4 + 23x^5 + 22x^6 + 15x^7 + 6x^8 +...
F(F(F(x))) = x + 6x^2 + (27)x^3 + 102x^4 + 333x^5 + 960x^6 + 2472x^7 +...
F(F(F(F(x)))) = x + 8x^2 + 52x^3 + (300)x^4 + 1578x^5 + 7692x^6 +...
F(F(F(F(F(x))))) = x + 10x^2 + 85x^3 + 660x^4 + (4790)x^5 + 32920x^6+...
F(F(F(F(F(F(x)))))) = x + 12x^2 +126x^3 +1230x^4+11385x^5+(101010)x^6+...

Cf. A119821, A112317, A119819, A119817, A119815.

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

a(n) = [x^n] F_n(x) where F_n(x) = F_{n-1}(F_1(x)) with F_1(x) = x*(1+x)^2.

A112319 Coefficients of x^n in the (n-1)-th iteration of (x + x^2) for n>=1.

Original entry on oeis.org

1, 1, 2, 9, 64, 630, 7916, 121023, 2179556, 45179508, 1059312264, 27715541568, 800423573676, 25289923553700, 867723362137464, 32128443862364255, 1276818947065793736, 54208515369076658640, 2448636361058495090816, 117254071399557173396416

Offset: 1

Paul D. Hanna, Sep 06 2005

nonn

			The iterations of (x+x^2) begin:
F(x) = x + (1)*x^2
F(F(x)) = x + 2*x^2 + (2)*x^3 + x^4
F(F(F(x))) = x + 3*x^2 + 6*x^3+ (9)*x^4 +...
F(F(F(F(x)))) = x + 4*x^2 + 12*x^3 + 30*x^4 + (64)*x^5 +...
F(F(F(F(F(x))))) = x + 5*x^2 + 20*x^3 + 70*x^4 + 220*x^5 + (630)*x^6 +...
coefficients enclosed in parenthesis form the initial terms of this sequence.

Paul D. Hanna, Table of n, a(n) for n = 1..200

Cf. A112317, A112320.

{a(n)=local(F=x+x^2, G=x+x*O(x^n));if(n<1,0, for(i=1,n-1,G=subst(F,x,G));return(polcoeff(G,n,x)))}
for(n=1,30,print1(a(n),", "))

a(n) = [x^n] F_{n-1}(x) where F_n(x) = F_{n-1}(x+x^2) with F_1(x) = x+x^2 and F_0(x)=x for n>=1.

A119821 Coefficients of x^n in the n-th iteration of x/(1-x)^2 for n>=1.

Original entry on oeis.org

1, 4, 33, 436, 8015, 189596, 5494797, 188692708, 7494744807, 338103170428, 17079035749061, 955117390512858, 58584586487137113, 3910851585418994256, 282272352712037938081, 21904366942822876046020

Offset: 1

Paul D. Hanna, Jun 01 2006

nonn

The coefficient of x^n in the n-th iteration of x/(1-x) = n^(n-1) = A000169(n); does this variant have a simple formula for a(n)?

			The successive iterations of F(x) = x/(1-x)^2 begin:
F(x) = (1)x + 2x^2 + 3x^3 + 4x^4 + 5x^5 + 6x^6 + 7x^7 + 8x^8 +...
F(F(x)) = x + (4)x^2 + 14x^3 + 46x^4 + 145x^5 + 444x^6 + 1331x^7 +...
F(F(F(x))) = x + 6x^2 + (33)x^3 + 174x^4 + 892x^5 + 4480x^6 +...
F(F(F(F(x)))) = x + 8x^2 + 60x^3 + (436)x^4 + 3102x^5 + 21728x^6 +...
F(F(F(F(F(x))))) = x + 10x^2 + 95x^3 + 880x^4 + (8015)x^5 +72090x^6+..
F(F(F(F(F(F(x)))))) = x + 12x^2+138x^3+1554x^4+17255x^5+(189596)x^6+..

Cf. A119820, A112317; A119819, A119817, A119815.

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

a(n) = [x^n] F_n(x) where F_n(x) = F_{n-1}(F(x)) with F(x) = x/(1-x)^2.

A119815 Integer a(n) produces the least positive integer coefficient of x^n in the n-th iteration of g.f. A(x) where A(0)=0.

Original entry on oeis.org

1, 1, -1, 1, 1, -11, 23, -20, 731, -4860, -91205, 138329, 24813133, 222203538, -11857627480, -340590475934, 7798573417057, 602467423292955, -4252676907049394, -1469602631093521547, -14928401886412967891, 4982240895059491727005, 167923794808862463264206

Offset: 1

Paul D. Hanna, May 31 2006

sign

			The iterated iterations of g.f. A(x) begin:
A(x) = (1)x + x^2 - x^3 + x^4 + x^5 - 11x^6 + 23x^7 - 20x^8 + 731x^9+..
A(A(x)) = x + (2)x^2 - 2x^4 + 6x^5 - 8x^6 - 50x^7 + 78x^8 + 1688x^9+...
A(A(A(x))) = x + 3x^2 + (3)x^3 - 3x^4 - x^5 + 17x^6 - 81x^7 -370x^8+...
A(A(A(A(x)))) = x + 4x^2 + 8x^3 + (4)x^4 - 12x^5 + 4x^6 + 12x^7 +...
A(A(A(A(A(x))))) = x + 5x^2 + 15x^3 + 25x^4 + (5)x^5 - 55x^6 -33x^7+...
A(A(A(A(A(A(x)))))) = x + 6x^2 + 24x^3 + 66x^4 + 106x^5 + (4)x^6 +...
Coefficients [x^n] of n-th self-composition of A(x) forms A119816:
[1,2,3,4,5,4,7,8,3,9,11,4,13,11,14,8,17,4,19,4,1,4,23,24,5,17,27,...].

Cf. A119816, A119817, A112317.

{a(n)=local(F=x+x^2+sum(k=3,n-1,a(k)*x^k),G=x+x*O(x^n));if(n<1,0,if(n<=2,1, for(k=1,n,G=subst(F,x,G));return((n-polcoeff(G,n,x)) )))}

A119817 Integer a(n) produces the least nonnegative integer coefficient of x^n in the n-th iteration of g.f. A(x).

Original entry on oeis.org

1, 1, -2, 8, -40, 210, -1032, 4074, -9084, -1485, -139344, -1178057, 97107644, 533077818, -43465435335, -997494915376, 35039558716800, 1885975569825115, -36684866143759995, -4946226556607087316, 24828007395162323458, 18213320246807011794109

Offset: 1

Paul D. Hanna, May 31 2006

sign

			The successive iterations of g.f. A(x) begin:
A(x) = (1)x + x^2 - 2x^3 + 8x^4 - 40x^5 + 210x^6 - 1032x^7 + 4074x^8+..
A(A(x)) = x + (2)x^2 - 2x^3 + 7x^4 - 30x^5 + 118x^6 -268x^7 -1430x^8+..
A(A(A(x))) = x + 3x^2 + (0)x^3 + 3x^4 -12x^5 +18x^6 +240x^7 -3119x^8+..
A(A(A(A(x)))) = x + 4x^2 + 4x^3 + (2)x^4 - 4x^5 - 18x^6 + 276x^7+...
A(A(A(A(A(x))))) = x + 5x^2 + 10x^3 + 10x^4 +(0)x^5 -20*x^6 +128*x^7+..
A(A(A(A(A(A(x)))))) = x + 6x^2 + 18x^3 +33x^4 +30x^5 +(0)x^6 -24x^7+..
Coefficients [x^n] of n-th iteration of A(x) forms A119818:
[1,2,0,2,0,0,0,0,0,0,0,10,0,0,7,12,0,6,0,9,2,11,0,8,10,13,18,18,...].

Cf. A119818, A119815, A112317.

{a(n)=local(F=x+x^2+sum(k=3,n-1,a(k)*x^k),G=x+x*O(x^n)); if(n<1,0,if(n<=2,1, for(k=1,n,G=subst(F,x,G)); return((n-1-polcoeff(G,n,x)) )))}

A119819 a(n) equals the coefficient of x^(n-1) in the (n-1)-th iteration of g.f. A(x) for n>1, with a(1)=1.

Original entry on oeis.org

1, 1, 2, 12, 138, 2370, 54190, 1553258, 53883088, 2211883428, 105760271082, 5819880201432, 364979361177134, 25865387272507770, 2056021496464455000, 182094050389241652004, 17861355920109599058260, 1929874166854161381238676, 228564755268775651632722308, 29540844190975459101114949972

Offset: 1

Paul D. Hanna, May 31 2006

nonn

Here the zeroth iteration of A(x) equals x, the first iteration is itself, the 2nd iteration of A(x) = A(A(x)), etc.

			The coefficients in the n-th iteration of g.f. A(x) begin:
n=1: [1, 1,  2,   12,   138,   2370,   54190,  1553258,   53883088, ...];
n=2: [1, 2,  6,   35,   370,   6000,  132344,  3704032,  126318024, ...];
n=3: [1, 3, 12,   75,   758,  11612,  245746,  6688885,  223699238, ...];
n=4: [1, 4, 20,  138,  1388,  20322,  411708, 10854152,  354952262, ...];
n=5: [1, 5, 30,  230,  2370,  33760,  656414, 16711414,  532707614, ...];
n=6: [1, 6, 42,  357,  3838,  54190, 1018484, 25016120,  775036254, ...];
n=7: [1, 7, 56,  525,  5950,  84630, 1553258, 36874397, 1107956996, ...];
n=8: [1, 8, 72,  740,  8888, 128972, 2337800, 53883088, 1568966580, ...];
n=9: [1, 9, 90, 1008, 12858, 192102, 3476622, 78308058, 2211883428, ...]; ...
where the diagonal of coefficients equals this sequence shift left 1 place.
...
More explicitly, the successive iterations of g.f. A(x) begin:
A(x) = (1)x + x^2 + 2x^3 + 12x^4 + 138x^5 + 2370x^6 + 54190x^7 +...
A(A(x)) = x + (2)x^2 + 6x^3 + 35x^4 + 370x^5 + 6000x^6 + 132344x^7 +...
A(A(A(x))) = x + 3x^2 + (12)x^3 + 75x^4 + 758x^5 + 11612x^6 +...
A(A(A(A(x)))) = x + 4x^2 + 20x^3 + (138)x^4 + 1388x^5 + 20322x^6 +...
A(A(A(A(A(x))))) = x + 5x^2 + 30x^3 + 230x^4 + (2370)x^5 + 33760x^6+...
A(A(A(A(A(A(x)))))) = x + 6x^2 +42x^3 +357x^4 +3838x^5 + (54190)x^6+...
...

Cf. A112317, A119820, A119821, A119815, A119817.

{a(n)=local(F=x+x^2+sum(m=3,n-1,a(m)*x^m), G=x+x*O(x^n)); if(n<1, 0,if(n<=2,1,for(i=1, n-1, G=subst(F, x, G)); return(polcoeff(G, n-1, x))))}
for(n=1,30,print1(a(n),", "))

A185755 Triangle: T(n,k) equals the coefficient of x^ny^k in the n-th iteration of x(1+xy)/(1-x), for n>=1, 0<=k

Original entry on oeis.org

1, 2, 2, 9, 15, 6, 64, 154, 120, 30, 625, 1995, 2340, 1190, 220, 7776, 31191, 49315, 38325, 14595, 2170, 117649, 571221, 1142932, 1204588, 704102, 215950, 27076, 2097152, 11992688, 29141994, 38972388, 30945432, 14570976, 3761310, 409836

Offset: 1

Paul D. Hanna, Feb 03 2011

			Triangle begins:
1;
2, 2;
9, 15, 6;
64, 154, 120, 30;
625, 1995, 2340, 1190, 220;
7776, 31191, 49315, 38325, 14595, 2170;
117649, 571221, 1142932, 1204588, 704102, 215950, 27076;
2097152, 11992688, 29141994, 38972388, 30945432, 14570976, 3761310, 409836;
43046721, 283976517, 814059798, 1323693384, 1334427720, 853356072, 337738758, 75550188, 7303164; ...

Cf. columns: A000169, A185756, A185757; row sums: A185523.

Cf. diagonals: A112317, A185758, A185759.

{T(n,k)=local(A=x, G=x*(1+x*y)/(1-x)); for(i=1, n, A=subst(G, x, A+x*O(x^n)));polcoeff(polcoeff(A, n,x),k,y)}

T(n,0) = A000169(n) = n^(n-1).
T(n,n) = A112317(n).
Sum_{k=0..n-1} T(n,k) = A185523(n).
Sum_{k=0..n-1} (-1)^k*T(n,k) = 0^n.

cp's OEIS Frontend

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A185755 Triangle: T(n,k) equals the coefficient of x^ny^k in the n-th iteration of x(1+xy)/(1-x), for n>=1, 0<=k