A177517 -id:A177517

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A186518

Triangle A177517 mod 2. Mahonian numbers modulo 2.

+20
2

1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

COMMENTS

Each column is a palindrome. The left half of the triangle displays chaos, the right half consists of triangles filled with zeros. Pattern is similar to patterns found in elementary cellular automata. The rule is different.

Compare the recurrence for this triangle:

T(1,1)=1, n > 1: T(n,1)=0, k > 1: T(n,k) = (Sum_{i=1..k-1} of T(n-i,k-1)) mod 2

with the recurrence for Dirichlet convolutions:

T(n,1)=1, k > 1: T(n,k) = ((Sum_{i=1..k-1} T(n-i,k-1)) mod 2) - ((Sum_{i=1..k-1} T(n-i,k)) mod 2) which in turn gives triangle A051731.

LINKS

G. C. Greubel, Rows n = 1..100 of triangle, flattened

Mats Granvik, Illustration of terms

FORMULA

T(1,1)=1, n > 1: T(n,1)=0, k > 1: T(n,k) = (Sum_{i=1..k-1} T(n-i,k-1)) mod 2.

EXAMPLE

Triangle starts:

0, 1;

0, 0, 1;

0, 0, 1, 1;

0, 0, 0, 0, 1;

0, 0, 0, 0, 1, 1;

0, 0, 0, 1, 1, 0, 1;

0, 0, 0, 0, 0, 1, 1, 1;

0, 0, 0, 0, 1, 1, 0, 0, 1;

0, 0, 0, 0, 1, 0, 1, 0, 1, 1;

0, 0, 0, 0, 1, 0, 1, 1, 1, 0, 1;

MAPLE

T:= proc(n, k) option remember;

if k=n then 1

else `mod`( add(T(n-j, k-1), j=1..k-1), 2)

fi; end:

seq(seq(T(n, k), k=1..n), n=1..12); # G. C. Greubel, Dec 17 2019

MATHEMATICA

nn = 19; t[n_, 1] = If[n==1, 1, 0]; t[n_, k_]:= t[n, k] = If[n>= k, Mod[Sum[t[n-i, k-1], {i, 1, k-1}], 2], 0]; Flatten[Table[t[n, k], {n, nn}, {k, n}]] (* Mats Granvik, Dec 06 2019 *)

PROG

(Excel cell formula, European) =if(column()=1; if(row()=1; 1; 0); if(row()>=column(); mod(sum(indirect(address(row()-column()+1; column()-1; 4)&":"&address(row()-1; column()-1; 4); 4)); 2); 0))

(Excel cell formula, American) =if(column()=1, if(row()=1, 1, 0), if(row()>=column(), mod(sum(indirect(address(row()-column()+1, column()-1, 4)&":"&address(row()-1, column()-1, 4), 4)), 2), 0))

(PARI) T(n, k) = if(k<1 || k>n, 0, if(k==n, 1, sum(j=1, k-1, T(n-j, k-1))%2 ));

for(n=1, 12, for(k=1, n, print1(T(n, k), ", "))) \\ G. C. Greubel, Dec 17 2019

(Magma)

function T(n, k)

if k lt 0 or k gt n then return 0;

elif k eq n then return 1;

elif k eq 1 then return 0;

else return (&+[T(n-j, k-1): j in [1..k-1]]) mod 2;

end if; return T; end function;

[T(n, k): k in [1..n], n in [1..12]]; // G. C. Greubel, Dec 17 2019

(Sage)

@CachedFunction

def T(n, k):

if (k<0 or k>n): return 0

elif (k==n): return 1

elif (k==1): return 0

else: return mod( sum(T(n-j, k-1) for j in (1..k-1)), 2)

[[T(n, k) for k in (1..n)] for n in (1..12)] # G. C. Greubel, Dec 17 2019

(GAP)

T:= function(n, k)

if k<0 or k>n then return 0;

elif k=n then return 1;

elif k=1 then return 0;

else return Sum([1..k-1], j-> T(n-j, k-1)) mod 2;

fi;

end;

Flat(List([1..12], n-> List([1..n], k-> T(n, k) ))); # G. C. Greubel, Dec 17 2019

CROSSREFS

Cf. A177517, A008302, A186519 (row sums).

KEYWORD

nonn,tabl

AUTHOR

Mats Granvik, Feb 23 2011

STATUS

approved

A186426

Antidiagonal sums of A177517.

+20
1

1, 0, 1, 0, 1, 1, 1, 2, 3, 4, 6, 10, 15, 23, 36, 57, 90, 142, 225, 358, 571, 912, 1458, 2334, 3742, 6006, 9648, 15510, 24951, 40164, 64689, 104241, 168048, 271015, 437221, 705571, 1138934, 1838916, 2969746, 4796900, 7749561, 12521629, 20235080, 32704173, 52862781, 85455587, 138156111, 223375201, 361186480, 584058718, 944511360, 1527499074, 2470446924, 3995662918, 6462775233, 10453566628, 16909223601, 27352387053, 44246403940, 71576559073

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,8

COMMENTS

a(n+1)/a(n) tends to the golden ratio. Grows more slowly than the Fibonacci sequence. More complicated than the Fibonacci sequence.

a(n)=First differences of A186425.

LINKS

Table of n, a(n) for n=1..60.

CROSSREFS

Cf. A001622, A177517, A186425.

KEYWORD

nonn

AUTHOR

Mats Granvik, Feb 21 2011

STATUS

approved

A177802

A row reversed version of A177517

+20
0

1, 1, 1, 0, 1, 1, 1, 2, 0, 1, 3, 2, 1, 4, 5, 1, 1, 5, 9, 6, 1, 6, 14, 15, 5, 1, 7, 20, 29, 20, 1, 8, 27, 49, 49, 22, 1, 9, 35, 76, 98, 71

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,8

COMMENTS

The row reversal allows getting rid of the initial zeros in A177517

and shows the polynomial structure of the sequence more clearly.

LINKS

Table of n, a(n) for n=1..42.

EXAMPLE

{1},

{1, 0},

{1, 1},

{1, 2, 0},

{1, 3, 2},

{1, 4, 5, 1},

{1, 5, 9, 6},

{1, 6, 14, 15, 5},

{1, 7, 20, 29, 20},

{1, 8, 27, 49, 49, 22},

{1, 9, 35, 76, 98, 71}

MATHEMATICA

Clear[t, n, k]

t[n_, 1] := If[n == 1, 1, 0]

t[n_, k_] := t[n, k] = Sum[t[n - i, k - 1], {i, 1, k - 1}]

a = Table[Reverse[Table[t[n, k], {k, 1, n}]], {n, 1, 12}];

Table[Table[a[[n, k]], {k, 1, Floor[(n + 1)/2]}], {n, 1, Length[a]}];

Flatten[%]

CROSSREFS

Cf. A177517

KEYWORD

nonn,tabf

AUTHOR

Roger L. Bagula and Mats Granvik, Dec 12 2010

STATUS

approved

A008930

Number of compositions (p_1, p_2, p_3, ...) of n with 1 <= p_i <= i for all i.

+10
36

1, 1, 1, 2, 3, 6, 11, 21, 41, 80, 157, 310, 614, 1218, 2421, 4819, 9602, 19147, 38204, 76266, 152307, 304256, 607941, 1214970, 2428482, 4854630, 9705518, 19405030, 38800412, 77585314, 155145677, 310251190, 620437691, 1240771141, 2481374234

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

COMMENTS

a(n) is the number of compositions (p_1,p_2,...) of n with 1<=p_i<=i for all i. a(n) is the number of Dyck n-paths with strictly increasing peaks. To get the correspondence, given such a Dyck path, split the path after the first up step reaching height i, i=1,2,...,h where h is the path's maximum height and count up steps in each block. Example: U-U-DUU-U-DDDD has been split as specified, yielding the composition (1,1,2,1). - David Callan, Feb 18 2004

Row sums of triangle A177517.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..3324 (first 201 terms from Vincenzo Librandi)

Margaret Archibald, Aubrey Blecher, Arnold Knopfmacher, and Stephan Wagner, Subdiagonal and superdiagonal compositions, Art Disc. Appl. Math. (2024). See p. 5.

Roland Bacher, Generic numerical semigroups, hal-03221466 [math.CO], 2021.

Sergi Elizalde, Symmetric peaks and symmetric valleys in Dyck paths, arXiv:2008.05669 [math.CO], 2020.

M. Torelli, Increasing integer sequences and Goldbach's conjecture, RAIRO - Theoretical Informatics and Applications, vol.40, no.02 (April 2006), pp.107-121.

FORMULA

G.f.: A(x) = Sum_{n>=0} x^n * Product_{k=1..n} (1-x^k)/(1-x). - Paul D. Hanna, Mar 20 2003

G.f.: A(x) = 1/(1 - x/(1+x) /(1 - x/(1+x+x^2) /(1 - x/(1+x+x^2+x^3) /(1 - x/(1+x+x^2+x^3+x^4) /(1 - x/(1+x+x^2+x^3+x^4+x^5) /(1 -...)))))), a continued fraction. - Paul D. Hanna, May 15 2012

Limit_{n->oo} a(n+1)/a(n) = 2. - Mats Granvik, Feb 22 2011

a(n) ~ c * 2^(n-1), where c = 0.288788095086602421... (see constant A048651). - Vaclav Kotesovec, Mar 17 2014

EXAMPLE

G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 3*x^4 + 6*x^5 + 11*x^6 + 21*x^7 + ...

The g.f. equals the following series involving q-factorials:

A(x) = 1 + x + x^2*(1+x) + x^3*(1+x)*(1+x+x^2) + x^4*(1+x)*(1+x+x^2)*(1+x+x^2+x^3) + x^5*(1+x)*(1+x+x^2)*(1+x+x^2+x^3)*(1+x+x^2+x^3+x^4) + ...

From Joerg Arndt, Dec 28 2012: (Start)

There are a(7)=21 compositions p(1)+p(2)+...+p(m) = 7 such that p(k) <= k:

[ 1] [ 1 1 1 1 1 1 1 ]

[ 2] [ 1 1 1 1 1 2 ]

[ 3] [ 1 1 1 1 2 1 ]

[ 4] [ 1 1 1 1 3 ]

[ 5] [ 1 1 1 2 1 1 ]

[ 6] [ 1 1 1 2 2 ]

[ 7] [ 1 1 1 3 1 ]

[ 8] [ 1 1 1 4 ]

[ 9] [ 1 1 2 1 1 1 ]

[10] [ 1 1 2 1 2 ]

[11] [ 1 1 2 2 1 ]

[12] [ 1 1 2 3 ]

[13] [ 1 1 3 1 1 ]

[14] [ 1 1 3 2 ]

[15] [ 1 2 1 1 1 1 ]

[16] [ 1 2 1 1 2 ]

[17] [ 1 2 1 2 1 ]

[18] [ 1 2 1 3 ]

[19] [ 1 2 2 1 1 ]

[20] [ 1 2 2 2 ]

[21] [ 1 2 3 1 ]

(End)

MAPLE

b:= proc(n, i) option remember; `if`(i>=n,

ceil(2^(n-1)), add(b(n-j, i+1), j=1..min(i, n)))

end:

a:= n-> b(n, 1):

seq(a(n), n=0..50); # Alois P. Heinz, Mar 25 2014, revised Jun 26 2023

MATHEMATICA

Sum[x^n*Product[(1-x^k)/(1-x), {k, 1, n}], {n, 0, 40}]+O[x]^41

Table[SeriesCoefficient[1+Sum[x^j*Product[(1-x^k)/(1-x), {k, 1, j}], {j, 1, n}], {x, 0, n}], {n, 0, 40}] (* Vaclav Kotesovec, Mar 17 2014 *)

b[n_, i_] := b[n, i] = If[n == 0, 1, Sum[b[n-j, i+1], {j, 1, Min[i, n]}]]; a[n_] := b[n, 1]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Apr 15 2015, after Alois P. Heinz *)

PROG

(PARI) { n=8; v=vector(n); for (i=1, n, v[i]=vector(i!)); v[1][1]=1; for (i=2, n, k=length(v[i-1]); for (j=1, k, for (a=0, i-1, v[i][j+a*k]=v[i-1][j]+a+1))); c=vector(n); for (i=1, n, for (j=1, i!, if (v[i][j]<=n, c[v[i][j]]++))); c } \\ Jon Perry

(PARI) N=66; q='q+O('q^N); Vec( sum(n=0, N, q^n * prod(k=1, n, (1-q^k)/(1-q) ) ) ) \\ Joerg Arndt, Mar 24 2014

CROSSREFS

Cf. A000707, A048285, A177517.

KEYWORD

nonn

AUTHOR

Mauro Torelli (torelli(AT)hermes.mc.dsi.unimi.it)

EXTENSIONS

More terms from Paul D. Hanna, Mar 20 2003

Offset corrected to 0 by Joerg Arndt, Mar 24 2014

New name (using comment by David Callan) from Joerg Arndt, Mar 25 2014

STATUS

approved

A177978

Triangle T(n,k) read by rows: A051731(n,k) - A051731(n-1,k).

+10
2

1, 0, 1, 0, -1, 1, 0, 1, -1, 1, 0, -1, 0, -1, 1, 0, 1, 1, 0, -1, 1, 0, -1, -1, 0, 0, -1, 1, 0, 1, 0, 1, 0, 0, -1, 1, 0, -1, 1, -1, 0, 0, 0, -1, 1, 0, 1, -1, 0, 1, 0, 0, 0, -1, 1, 0, -1, 0, 0, -1, 0, 0, 0, 0, -1, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0, -1, 1, 0, -1, -1, -1, 0, -1, 0, 0, 0, 0, 0, -1, 1, 0, 1

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

The recurrence for this triangle is similar to the recurrence in A177517. Cumulative column sums give table A051731.

LINKS

Table of n, a(n) for n=1..93.

FORMULA

T(n,1)=A000007, k > 1: T(n,k) = (Sum_{i=1..k-1} T(n-i,k-1)) - (Sum_{i=1..k-1} T(n-i,k)).

EXAMPLE

Triangle begins:

0, 1;

0, -1, 1;

0, 1, -1, 1;

0, -1, 0, -1, 1;

0, 1, 1, 0, -1, 1;

0, -1, -1, 0, 0, -1, 1;

0, 1, 0, 1, 0, 0, -1, 1;

0, -1, 1, -1, 0, 0, 0, -1, 1;

0, 1, -1, 0, 1, 0, 0, 0, -1, 1;

0, -1, 0, 0, -1, 0, 0, 0, 0, -1, 1;

0, 1, 1, 1, 0, 1, 0, 0, 0, 0, -1, 1;

0, -1, -1, -1, 0, -1, 0, 0, 0, 0, 0, -1, 1;

0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, -1, 1;

PROG

(Excel cell formula, American) =if(column()=1, if(row()>1, 0, 1), if(row()>=column(), sum(indirect(address(row()-column()+1, column()-1, 4)&":"&address(row()-1, column()-1, 4), 4))-sum(indirect(address(row()-column()+1, column(), 4)&":"&address(row()-1, column(), 4), 4)), 0))

(Excel cell formula, European)=if(column()=1; if(row()>1; 0; 1); if(row()>=column(); sum(indirect(address(row()-column()+1; column()-1; 4)&":"&address(row()-1; column()-1; 4); 4))-sum(indirect(address(row()-column()+1; column(); 4)&":"&address(row()-1; column(); 4); 4)); 0))

CROSSREFS

Matrix inverse of A134540. Cf. A177517.

KEYWORD

sign,tabl

AUTHOR

Mats Granvik, May 16 2010

EXTENSIONS

Typo in sequence (erroneous comma) corrected by N. J. A. Sloane, May 22 2010

Edited by Mats Granvik, Dec 11 2010

STATUS

approved

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