A091768 -id:A091768 - cp's OEIS frontend

A163946 Triangle read by rows, A033184 * A091768 (diagonalized as an infinite lower triangular matrix).

1, 1, 1, 2, 2, 2, 5, 5, 6, 6, 14, 14, 18, 24, 22, 42, 42, 56, 84, 110, 92, 132, 132, 180, 288, 440, 552, 426, 429, 429, 594, 990, 1650, 2484, 2982, 2150, 1430, 1430, 2002, 3432, 6050, 10120, 14910, 17200, 11708, 4862, 4862, 6864, 12012, 22022, 39468, 65604, 94600, 105372, 68282

Offset: 0

Gary W. Adamson, Aug 06 2009

As an eigentriangle, equals A033184 * the diagonalized version of its eigensequence. (the eigensequence of triangle A033184 = A091768).
Right border = A091768, left border = Catalan sequence A000108.
Sum of n-th row terms = rightmost term of next row.

			First few rows of the triangle =
  1;
  1, 1;
  2, 2, 2;
  5, 5, 6, 6;
  14, 14, 18, 24, 22;
  42, 42, 56, 84, 110, 92;
  132, 132, 180, 288, 440, 552, 426;
  429, 429, 594, 990, 1650, 2484, 2982, 2150;
  1430, 1430, 2002, 3432, 6050, 10120, 14910, 17200, 11708;
  4862, 4862, 6864, 12012, 22022, 39468, 65604, 94600, 105372, 68282;
  ...
Row 3 = (5, 5, 6, 6) = (5, 5, 3, 1) * (1, 1, 2, 6); where (5, 5, 3, 1) = row 3 of triangle A033184 and (1, 1, 2, 6) = the first 3 terms of A091768 prefaced with a 1.

Cf. A033184, A091768, A000108, A071721.

Triangle read by rows, A033184 * A091768 (diagonalized such that the right border = (1, 1, 2, 6, 22, 92, 426, 2150,...) i.e. A091768 prefaced with a 1; with the rest zeros).

A336070 Number of inversion sequences avoiding the vincular pattern 10-0 (or 10-1).

Original entry on oeis.org

1, 1, 2, 6, 23, 106, 567, 3440, 23286, 173704, 1414102, 12465119, 118205428, 1199306902, 12958274048, 148502304614, 1798680392716, 22953847041950, 307774885768354, 4325220458515307, 63563589415836532, 974883257009308933, 15575374626562632462, 258780875395778033769, 4464364292401926006220

Offset: 0

Michael De Vlieger, Jul 07 2020

nonn

From Joerg Arndt, Jan 20 2024: (Start)
a(n) is the number of weak ascent sequences of length n.
A weak ascent sequence is a sequence [d(1), d(2), ..., d(n)] where d(1)=0, d(k)>=0, and d(k) <= 1 + asc([d(1), d(2), ..., d(k-1)]) and asc(.) counts the weak ascents d(j) >= d(j-1) of its argument.
The number of length-n weak ascent sequences with maximal number of weak ascents is A000108(n).
(End)

			From _Joerg Arndt_, Jan 20 2024: (Start)
There are a(4) = 23 weak ascent sequences (dots for zeros):
   1:  [ . . . . ]
   2:  [ . . . 1 ]
   3:  [ . . . 2 ]
   4:  [ . . . 3 ]
   5:  [ . . 1 . ]
   6:  [ . . 1 1 ]
   7:  [ . . 1 2 ]
   8:  [ . . 1 3 ]
   9:  [ . . 2 . ]
  10:  [ . . 2 1 ]
  11:  [ . . 2 2 ]
  12:  [ . . 2 3 ]
  13:  [ . 1 . . ]
  14:  [ . 1 . 1 ]
  15:  [ . 1 . 2 ]
  16:  [ . 1 1 . ]
  17:  [ . 1 1 1 ]
  18:  [ . 1 1 2 ]
  19:  [ . 1 1 3 ]
  20:  [ . 1 2 . ]
  21:  [ . 1 2 1 ]
  22:  [ . 1 2 2 ]
  23:  [ . 1 2 3 ]
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..400
Juan S. Auli and Sergi Elizalde, Wilf equivalences between vincular patterns in inversion sequences, arXiv:2003.11533 [math.CO], 2020. See p. 5, Table 1. Gives terms 1-10.
Beata Benyi, Anders Claesson, and Mark Dukes, Weak ascent sequences and related combinatorial structures, arXiv:2111.03159 [math.CO], (4-November-2021).

Cf. A000079, A000108, A000110, A022493, A091768, A102038, A113227, A263777, A328441, A336071, A336072.

Row sums of A369321.

b:= proc(n, i, t) option remember; `if`(n=0, 1,
      add(b(n-1, j, t+`if`(j>=i, 1, 0)), j=0..t+1))
    end:
a:= n-> b(n, -1$2):
seq(a(n), n=0..25);  # Alois P. Heinz, Jan 23 2024

b[n_, i_, t_] := b[n, i, t] = If[n == 0, 1, Sum[b[n - 1, j, t + If[j >= i, 1, 0]], {j, 0, t + 1}]];
a[n_] := b[n, -1, -1];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Jan 18 2025, after Alois P. Heinz *)

\\ see formula (5) on page 18 of the Benyi/Claesson/Dukes reference
N=40;
M=matrix(N,N,r,c,-1);  \\ memoization
a(n,k)=
{
    if ( n==0 && k==0, return(1) );
    if ( k==0, return(0) );
    if ( n==0, return(0) );
    if ( M[n,k] != -1 , return( M[n,k] ) );
    my( s );
    s = sum( i=0, n, sum( j=0, k-1,
         (-1)^j * binomial(k-j,i) * binomial(i,j) * a( n-i, k-j-1 )) );
    M[n,k] = s;
    return( s );
}
for (n=0, N, print1( sum(k=1,n,a(n,k)),", "); );
\\ print triangle a(n,k), see A369321:
\\ for (n=0, N, for(k=0,n, print1(a(n,k),", "); ); print(););
\\ Joerg Arndt, Jan 20 2024

a(0)=1 prepended and more terms from Joerg Arndt, Jan 20 2024

A172380 Eigentriangle of Catalan triangle A033184.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 3, 1, 0, 5, 10, 6, 1, 0, 14, 36, 31, 10, 1, 0, 42, 137, 156, 75, 15, 1, 0, 132, 544, 787, 510, 155, 21, 1, 0, 429, 2235, 4017, 3331, 1380, 287, 28, 1, 0, 1430, 9445, 20809, 21405, 11411, 3255, 490, 36, 1, 0, 4862, 40876, 109486, 136921, 90665

Offset: 0

Paul Barry, Feb 01 2010

Row sums are A091768.
Production matrix of inverse is matrix with general term (-1)^(n-k+1)C(k,n-k+1).
Diagonal sums are A172382. Product of A033184 and A172380 is the matrix A172381.

			Triangle begins
  1;
  0,    1;
  0,    1,    1;
  0,    2,    3,     1;
  0,    5,   10,     6,     1;
  0,   14,   36,    31,    10,     1;
  0,   42,  137,   156,    75,    15,    1;
  0,  132,  544,   787,   510,   155,   21,   1;
  0,  429, 2235,  4017,  3331,  1380,  287,  28,  1;
  0, 1430, 9445, 20809, 21405, 11411, 3255, 490, 36, 1;
Production matrix of inverse is
  0,  1;
  0, -1,  1;
  0,  0, -2,  1;
  0,  0,  1, -3,  1;
  0,  0,  0,  3, -4,   1;
  0,  0,  0, -1,  6,  -5,   1;
  0,  0,  0,  0, -4,  10,  -6,   1;
  0,  0,  0,  0,  1, -10,  15,  -7,   1;
  0,  0,  0,  0,  0,   5, -20,  21,  -8,  1;
  0,  0,  0,  0,  0,  -1,  15, -35,  28, -9,   1;
  0,  0,  0,  0,  0,   0,  -6,  35, -56, 36, -10, 1;

Paul Barry, Invariant number triangles, eigentriangles and Somos-4 sequences, arXiv preprint arXiv:1107.5490 [math.CO], 2011.

A336071 Number of inversion sequences avoiding the vincular pattern 1-01 (or 1-10).

Original entry on oeis.org

1, 2, 6, 23, 107, 584, 3655, 25790, 202495, 1750763

Offset: 1

Michael De Vlieger, Jul 07 2020

Juan S. Auli, Sergi Elizalde, Wilf equivalences between vincular patterns in inversion sequences, arXiv:2003.11533 [math.CO], 2020. See p. 5, Table 1. Gives terms 1-10.

Cf. A000079, A000110, A022493, A091768, A102038, A113227, A263777, A328441, A336070, A336072.

A336072 Number of inversion sequences avoiding the vincular pattern 2-01 (or 2-10).

Original entry on oeis.org

1, 2, 6, 24, 118, 680, 4460, 32634, 262536, 2296532

Offset: 1

Michael De Vlieger, Jul 07 2020

Juan S. Auli, Sergi Elizalde, Wilf equivalences between vincular patterns in inversion sequences, arXiv:2003.11533 [math.CO], 2020. See p. 5, Table 1. Gives terms 1-10.

Cf. A000079, A000110, A022493, A091768, A102038, A113227, A263777, A328441, A336070, A336071.

A181644 Eigentriangle for the Catalan triangle (c(x), xc(x)), A033184.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 6, 3, 1, 1, 22, 11, 4, 1, 1, 92, 46, 17, 5, 1, 1, 426, 213, 79, 24, 6, 1, 1, 2150, 1075, 399, 122, 32, 7, 1, 1, 11708, 5854, 2173, 665, 176, 41, 8, 1, 1, 68282, 34141, 12673, 3878, 1027, 242, 51, 9, 1, 1, 423948, 211974, 78683, 24075, 6373, 1502

Offset: 0

Paul Barry, Nov 03 2010

First column is essentially A091768. Inverse of A181645.

			Triangle begins
  1,
  1, 1,
  2, 1, 1,
  6, 3, 1, 1,
  22, 11, 4, 1, 1,
  92, 46, 17, 5, 1, 1,
  426, 213, 79, 24, 6, 1, 1,
  2150, 1075, 399, 122, 32, 7, 1, 1,
  11708, 5854, 2173, 665, 176, 41, 8, 1, 1

Paul Barry, Invariant number triangles, eigentriangles and Somos-4 sequences, arXiv preprint arXiv:1107.5490 [math.CO], 2011.

Cf. A033184, A091768, A181645.

A125277 Eigensequence of triangle A110616: a(n) = Sum_{k=0..n-1} A110616(n-1,k)*a(k) for n>0 with a(0)=1.

Original entry on oeis.org

1, 1, 2, 7, 32, 169, 981, 6113, 40386, 280871, 2047316, 15595317, 123876270, 1024188790, 8799533250, 78443220865, 724472766347, 6922133112818, 68331103658847, 695983854400857, 7305630631254242, 78941171881894716

Offset: 0

Paul D. Hanna, Nov 26 2006

nonn

			a(3) = 3*(1) + 2*(1) + 1*(2) = 7;
a(4) = 12*(1) + 7*(1) + 3*(2) + 1*(7) = 32;
a(5) = 55*(1) + 30*(1) + 12*(2) + 4*(7) + 1*(32) = 169.
Triangle A110616(n,k) = C(3*n-2*k+1, n-k)*(k+1)/(3*n-2*k+1) begins:
1;
1, 1;
3, 2, 1;
12, 7, 3, 1;
55, 30, 12, 4, 1;
273, 143, 55, 18, 5, 1;
1428, 728, 273, 88, 25, 6, 1; ...
where g.f. of column k = G001764(x)^(k+1)
and G001764(x) = 1 + x*G001764(x)^3 is the g.f. of A001764.

Cf. A110616, A001764, A091768.

{a(n)=if(n==0,1,sum(k=0,n-1, a(k)*binomial(3*n-2*k-2,n-k-1)*(k+1)/(3*n-2*k-2)))}

a(n) = Sum_{k=0..n-1} a(k) * C(3*n-2*k-2,n-k-1)*(k+1)/(3*n-2*k-2) for n>0 with a(0)=1.

cp's OEIS Frontend

A163946 Triangle read by rows, A033184 * A091768 (diagonalized as an infinite lower triangular matrix).

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A336070 Number of inversion sequences avoiding the vincular pattern 10-0 (or 10-1).

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A172380 Eigentriangle of Catalan triangle A033184.

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A336071 Number of inversion sequences avoiding the vincular pattern 1-01 (or 1-10).

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A336072 Number of inversion sequences avoiding the vincular pattern 2-01 (or 2-10).

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A181644 Eigentriangle for the Catalan triangle (c(x), xc(x)), A033184.

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A125277 Eigensequence of triangle A110616: a(n) = Sum_{k=0..n-1} A110616(n-1,k)*a(k) for n>0 with a(0)=1.

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