A166900 -id:A166900 - cp's OEIS frontend

A166901 Column 1 of triangle A166900.

1, 4, 21, 156, 1540, 19160, 288813, 5123608, 104657520, 2420186616, 62514944778, 1784255891484, 55767065855228, 1894463658611680, 69504774168222109, 2738952451360200312, 115380142451625516088, 5174227834995200591840

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, A166902, A166903, A122888, A135080.

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

A166902 Column 2 of triangle A166900.

Original entry on oeis.org

1, 9, 84, 935, 12480, 196623, 3591560, 74847168, 1755406674, 45804773872, 1317004696656, 41386864224420, 1411592788770580, 51942256939923051, 2051313029747633376, 86548588478842559964, 3885584044838123386104

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, A166903, A122888, A135080.

{a(n)=local(F=x, M, N, P); M=matrix(n+4, n+4, r, c, F=x;for(i=1, r+c-2, F=subst(F, x, x+x^2+x*O(x^(n+4)))); polcoeff(F, c)); N=matrix(n+3, n+3, r, c, F=x;for(i=1, r, F=subst(F, x, x+x^2+x*O(x^(n+4)))); polcoeff(F, c)); P=matrix(n+3, n+3, r, c, M[r+1, c]); (P~*N~^-1)[n+3, 3]}

A166903 Column 3 of triangle A166900.

Original entry on oeis.org

1, 16, 230, 3564, 61845, 1207696, 26415840, 642448632, 17240108314, 506777596248, 16210958231104, 560988459704192, 20891752852722701, 833382707754108896, 35461362393617267808, 1603581518693484768464

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, A166904, A122888, A135080.

{a(n)=local(F=x, M, N, P); M=matrix(n+5, n+5, r, c, F=x;for(i=1, r+c-2, F=subst(F, x, x+x^2+x*O(x^(n+5)))); polcoeff(F, c)); N=matrix(n+4, n+4, r, c, F=x;for(i=1, r, F=subst(F, x, x+x^2+x*O(x^(n+5)))); polcoeff(F, c)); P=matrix(n+4, n+4, r, c, M[r+1, c]); (P~*N~^-1)[n+4, 4]}

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])}

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

A112317 Coefficients of x^n in the n-th iteration of (x + x^2) for n>=1.

Original entry on oeis.org

1, 2, 6, 30, 220, 2170, 27076, 409836, 7303164, 149837028, 3479498880, 90230486346, 2584679465160, 81056989408928, 2762187020749144, 101633218030586364, 4015771398425994048, 169588657820702174728

Offset: 1

Paul D. Hanna, Sep 03 2005

nonn

Forms a diagonal of the tables A122888 and A185755.

			The initial iterations of x + x^2 begin:
F(x) = (1)*x + 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+ 10*x^5+ 8*x^6+ 4*x^7+ x^8;
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 +...;
F(F(F(F(F(F(x)))))) = x + 6*x^2 + 30*x^3 + 135*x^4 + 560*x^5 + (2170)*x^6 +...;
where the terms in parenthesis illustrate how to form this sequence.

Paul D. Hanna and Vaclav Kotesovec, Table of n, a(n) for n = 1..300 (first 100 terms from Paul D. Hanna)

Cf. A112319, A112320, A122888, A185755, A135080, A166900.

{a(n)=local(F=x+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)))}
for(n=1, 30, print1(a(n), ", "))

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

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

A071207 Triangular array T(n,k) read by rows, giving number of rooted trees on the vertex set {1..n+1} where k children of the root have a label smaller than the label of the root.

Original entry on oeis.org

1, 1, 1, 4, 4, 1, 27, 27, 9, 1, 256, 256, 96, 16, 1, 3125, 3125, 1250, 250, 25, 1, 46656, 46656, 19440, 4320, 540, 36, 1, 823543, 823543, 352947, 84035, 12005, 1029, 49, 1, 16777216, 16777216, 7340032, 1835008, 286720, 28672, 1792, 64, 1, 387420489

Offset: 0

Cedric Chauve (chauve(AT)lacim.uqam.ca), May 16 2002

The n-th term of the n-th binomial transform of a sequence {b} is given by {d} where d(n) = sum(k=0,n,T(n,k)*b(k)) and T(n,k)=binomial(n,k)*n^(n-k); such diagonals are related to the hyperbinomial transform (A088956). - Paul D. Hanna, Nov 04 2003
T(n,k) gives the number of divisors of A181555(n) with (n-k) distinct prime factors. See also A001221, A146289, A146290, A181567. - Matthew Vandermast, Oct 31 2010
T(n,k) is the number of partial functions on {1,2,...,n} leaving exactly k elements undefined.  Row sums = A000169. - Geoffrey Critzer, Jan 08 2012
As a triangular matrix, transforms rows into diagonals in the table of coefficients of successive iterations of x/(1-x). - Paul D. Hanna, Jan 19 2014
Also the number of rooted trees on n+1 labeled vertices in which some specified vertex (say, vertex 1) has k children. - Alan Sokal, Jul 22 2022

			1
1     1
4     4     1
27    27    9     1
256   256   96    16    1
3125  3125  1250  250   25    1
46656 46656 19440 4320  540   36    1

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
C. Chauve, S. Dulucq and O. Guibert, Enumeration of some labeled trees, Proceedings of FPSAC/SFCA 2000 (Moscow), Springer, pp. 146-157.
Alan D. Sokal, A remark on the enumeration of rooted labeled trees, arXiv:1910.14519 [math.CO], 2019 and Discrete Math. 343, 111865 (2020).

Cf. A000169, A000312, A089466, A088956, A166900.

T:= (n, k)-> binomial(n, k)*n^(n-k): seq(seq(T(n, k), k=0..n), n=0..10);

Prepend[Flatten[ Table[Table[Binomial[n, k] n^(n - k), {k, 0, n}], {n, 1, 8}]], 1]  (* Geoffrey Critzer, Jan 08 2012 *)

T(n,k)=if(k<0 || k>n,0,binomial(n,k)*n^(n-k))

/* Transforms rows into diagonals in the iterations of x/(1-x): */
{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/(1-x+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/(1-x+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("")) \\ Paul D. Hanna, Jan 19 2014

T(n,k) = binomial(n, k)*n^(n-k).

E.g.f.: (-LambertW(-y)/y)^x/(1+LambertW(-y)). - Vladeta Jovovic

Name edited by Alan Sokal, Jul 22 2022

A166905 Triangle, read by rows, that transforms rows into diagonals in the table A158825 of coefficients in successive iterations of x*Catalan(x) (cf. A000108).

Original entry on oeis.org

1, 1, 1, 6, 4, 1, 54, 33, 9, 1, 640, 380, 108, 16, 1, 9380, 5510, 1610, 270, 25, 1, 163576, 95732, 28560, 5148, 570, 36, 1, 3305484, 1933288, 586320, 110929, 13650, 1071, 49, 1, 75915708, 44437080, 13658904, 2677008, 353600, 31624, 1848, 64, 1, 1952409954

Offset: 0

Paul D. Hanna, Nov 28 2009

			Triangle begins:
1;
1,1;
6,4,1;
54,33,9,1;
640,380,108,16,1;
9380,5510,1610,270,25,1;
163576,95732,28560,5148,570,36,1;
3305484,1933288,586320,110929,13650,1071,49,1;
75915708,44437080,13658904,2677008,353600,31624,1848,64,1;
1952409954,1144564278,355787568,71648322,9962949,973845,66150,2988,81,1;
55573310936,32638644236,10243342296,2107966432,304857190,31795560,2395120,127720,4590,100,1;
...
Coefficients in iterations of x*Catalan(x) form table A158825:
1,1,2,5,14,42,132,429,1430,4862,16796,58786,208012,742900,...;
1,2,6,21,80,322,1348,5814,25674,115566,528528,2449746,...;
1,3,12,54,260,1310,6824,36478,199094,1105478,6227712,...;
1,4,20,110,640,3870,24084,153306,993978,6544242,43652340,...;
1,5,30,195,1330,9380,67844,500619,3755156,28558484,...;
1,6,42,315,2464,19852,163576,1372196,11682348,100707972,...;
1,7,56,476,4200,38052,351792,3305484,31478628,303208212,...;
...
This triangle T transforms rows into diagonals of A158825;
the initial diagonals begin:
A158831: [1,1,6,54,640,9380,163576,3305484,...];
A158832: [1,2,12,110,1330,19852,351792,7209036,...];
A158833: [1,3,20,195,2464,38052,693048,14528217,...];
A158834: [1,4,30,315,4200,67620,1273668,27454218,...].
For example:
T * [1,0,0,0,0,0,0,0,0,0,0,0,0, ...] = A158831;
T * [1,1,2,5,14,42,132,429,1430,...] = A158832;
T * [1,2,6,21,80,322,1348,5814, ...] = A158833;
T * [1,3,12,54,260, 1310, 6824, ...] = A158834.

Cf. A166906, A166907, A166908, A166909, variant: A166900.

Cf. A158825, A158831, A158832, A158833, A158834, A158835, A000108.

{T(n, k)=local(F=x, G=serreverse(x-x^2+O(x^(n+3))), 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, G+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, G+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]}

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A166901 Column 1 of triangle A166900.

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A166902 Column 2 of triangle A166900.

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A166903 Column 3 of triangle A166900.

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A166904 Row sums of triangle A166900.

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

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A112317 Coefficients of x^n in the n-th iteration of (x + x^2) for n>=1.

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A071207 Triangular array T(n,k) read by rows, giving number of rooted trees on the vertex set {1..n+1} where k children of the root have a label smaller than the label of the root.

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A166905 Triangle, read by rows, that transforms rows into diagonals in the table A158825 of coefficients in successive iterations of x*Catalan(x) (cf. A000108).

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