A304190 -id:A304190 - cp's OEIS frontend

A132615 Triangle T, read by rows, where row n+1 of T = row n of T^(2n-1) with appended '1' for n>=0 with T(0,0)=1.

1, 1, 1, 1, 1, 1, 6, 3, 1, 1, 80, 25, 5, 1, 1, 1666, 378, 56, 7, 1, 1, 47232, 8460, 1020, 99, 9, 1, 1, 1694704, 252087, 26015, 2134, 154, 11, 1, 1, 73552752, 9392890, 855478, 61919, 3848, 221, 13, 1, 1, 3744491970, 420142350, 34461260, 2257413, 125760, 6290, 300, 15, 1, 1

Offset: 0

Paul D. Hanna, Aug 24 2007

			Triangle begins:
  1;
  1, 1;
  1, 1, 1;
  6, 3, 1, 1;
  80, 25, 5, 1, 1;
  1666, 378, 56, 7, 1, 1;
  47232, 8460, 1020, 99, 9, 1, 1;
  1694704, 252087, 26015, 2134, 154, 11, 1, 1;
  73552752, 9392890, 855478, 61919, 3848, 221, 13, 1, 1; ...
GENERATE T FROM ODD MATRIX POWERS OF T.
Matrix cube, T^3, begins:
  1;
  3, 1;
  6, 3, 1; <-- row 3 of T
  31, 12, 3, 1;
  357, 100, 18, 3, 1;
  6786, 1455, 205, 24, 3, 1; ...
where row 3 of T = row 2 of T^3 with appended '1'.
Matrix fifth power, T^5, begins:
  1;
  5, 1;
  15, 5, 1;
  80, 25, 5, 1; <-- row 4 of T
  855, 215, 35, 5, 1;
  15171, 3065, 410, 45, 5, 1; ...
where row 4 of T = row 3 of T^5 with appended '1'.
Matrix seventh power, T^7, begins:
  1;
  7, 1;
  28, 7, 1;
  161, 42, 7, 1;
  1666, 378, 56, 7, 1; <-- row 5 of T
  28119, 5348, 679, 70, 7, 1; ...
where row 5 of T = row 4 of T^7 with appended '1'.
ALTERNATE GENERATING METHOD.
Row 4: start with a '1' followed by 4 zeros;
take partial sums and append 2 zeros; then
take partial sums thrice more:
  (1), 0, 0, 0, 0;
  1, 1, 1, 1, (1), 0, 0;
  1, 2, 3, 4, 5, 5, (5);
  1, 3, 6, 10, 15, 20, (25);
  1, 4, 10, 20, 35, 55, (80);
the final nonzero terms form row 4: [80, 25, 5, 1, 1].
Row 5: start with a '1' followed by 6 zeros;
take partial sums and append 4 zeros;
take partial sums and append 2 zeros; then
take partial sums thrice more:
  (1), 0, 0, 0, 0, 0, 0;
  1, 1, 1, 1, 1, 1, (1), 0, 0, 0, 0;
  1, 2, 3, 4, 5, 6, 7, 7, 7, 7, (7), 0, 0;
  1, 3, 6, 10, 15, 21, 28, 35, 42, 49, 56, 56, (56);
  1, 4, 10, 20, 35, 56, 84, 119, 161, 210, 266, 322, (378);
  1, 5, 15, 35, 70, 126, 210, 329, 490, 700, 966, 1288, (1666);
the final nonzero terms form row 5: [1666, 378, 56, 7, 1, 1].
Continuing in this way produces all the rows of this triangle.

Alois P. Heinz, Rows n = 0..140, flattened

Cf. columns: A132616, A132617, A132618; A132619; variants: A132610, A101479.

Cf. A304190, A304191, A304193.

b:= proc(n) option remember;
      Matrix(n, (i,j)-> T(i-1,j-1))^(2*n-3)
    end:
T:= proc(n,k) option remember;
     `if`(n=k, 1, `if`(k>n, 0, b(n)[n,k+1]))
    end:
seq(seq(T(n,k), k=0..n), n=0..10);  # Alois P. Heinz, Apr 13 2020

b[n_] := b[n] = MatrixPower[Table[T[i-1, j-1], {i, n}, {j, n}], 2n-3];
T[n_, k_] := T[n, k] = If[n == k, 1, If[k > n, 0, b[n][[n, k + 1]]]];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 27 2020, after Alois P. Heinz *)

{T(n, k)=local(A=vector(n+1), p); A[1]=1; for(j=1, n-k-1, p=(n-1)*(n-2)-(n-j-1)*(n-j-2); A=Vec((Polrev(A)+x*O(x^p))/(1-x))); A=Vec((Polrev(A)+x*O(x^p))/(1-x)); A[p+1]}

T(n+1,1) is divisible by 2n-1 for n>=1.

A304193 G.f. A(x) satisfies: [x^n] (1+x)^((n+1)^2) / A(x) = 0 for n>0.

Original entry on oeis.org

1, 4, 16, 144, 2346, 55236, 1688084, 63040736, 2770165274, 139623836116, 7925496107656, 499719554537584, 34625595715906866, 2613946666882042164, 213475621178226876156, 18748792440158256161216, 1761875767691411063734514, 176383456081424163875684516, 18739798321516251204837796864, 2105891800817103192582808107856

Offset: 0

Paul D. Hanna, May 07 2018

nonn

Note that: [x^n] (1+x)^((n+1)*k) / G(x) = 0 for n>0 holds when G(x) = (1+x)^(k+1)/(1 - (k-1)*x) given some fixed k ; this sequence explores the case where k varies with n.

			G.f.: A(x) = 1 + 4*x + 16*x^2 + 144*x^3 + 2346*x^4 + 55236*x^5 + 1688084*x^6 + 63040736*x^7 + 2770165274*x^8 + 139623836116*x^9 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in (1+x)^((n+1)^2) / A(x) begins:
n=0: [1, -3, -4, -80, -1530, -40222, -1316104, -51439572, ...];
n=1: [1, 0, -10, -100, -1785, -45056, -1441440, -55510080, ...];
n=2: [1, 5, 0, -140, -2380, -55080, -1685620, -63186200, ...];
n=3: [1, 12, 56, 0, -3150, -74484, -2125948, -76230384, ...];
n=4: [1, 21, 200, 1020, 0, -96492, -2901052, -98301840, ...];
n=5: [1, 32, 486, 4540, 26015, 0, -3718000, -135081440, ...];
n=6: [1, 45, 980, 13640, 132810, 855478, 0, -172046940, ...];
n=7: [1, 60, 1760, 33520, 462150, 4790156, 34461260, 0, ...]; ...
in which the main diagonal is all zeros after the initial term, illustrating that [x^n] (1+x)^((n+1)^2) / A(x) = 0 for n>0.
RELATED SEQUENCES.
The secondary diagonal in the above table that begins
[1, 5, 56, 1020, 26015, 855478, 34461260, 1642995124, ...]
yields A132618, column 2 of triangle A132615.
Related triangular matrix T = A132615 begins:
1;
1, 1;
1, 1, 1;
6, 3, 1, 1;
80, 25, 5, 1, 1;
1666, 378, 56, 7, 1, 1;
47232, 8460, 1020, 99, 9, 1, 1;
1694704, 252087, 26015, 2134, 154, 11, 1, 1;
73552752, 9392890, 855478, 61919, 3848, 221, 13, 1, 1; ...
in which row n equals row (n-1) of T^(2*n-1) followed by '1' for n > 0.

Paul D. Hanna, Table of n, a(n) for n = 0..300

Cf. A132618, A304190, A304191, A304192, A132615.

{a(n) = my(A=[1]); for(i=1,n, A=concat(A,0); m=#A; A[m] = Vec( (1+x +x*O(x^m))^(m^2)/Ser(A) )[m] ); A[n+1]}
for(n=0,30, print1(a(n),", "))

A132618(n+1) = [x^n] (1+x)^((n+2)^2) / A(x) for n>=0.

A304191 G.f. A(x) satisfies: [x^n] (1+x)^(n^2) / A(x) = 0 for n > 0.

Original entry on oeis.org

1, 1, 3, 35, 611, 14691, 448873, 16606825, 720241161, 35786093321, 2002505540123, 124546575282555, 8520012343770331, 635618668572015451, 51348334729127568273, 4465119223213849398545, 415808496978034659793361, 41283870149540066960271441, 4353184675864365012327673843, 485828603554439779231472806675

Offset: 0

Paul D. Hanna, May 07 2018

nonn

Note that [x^n] (1+x)^(n*k) / G(x) = 0 for n > 0 holds when G(x) = (1+x)/(1 - (k-1)*x) given some fixed k; this sequence explores the case where k varies with n.

			G.f.: A(x) = 1 + x + 3*x^2 + 35*x^3 + 611*x^4 + 14691*x^5 + 448873*x^6 + 16606825*x^7 + 720241161*x^8 + 35786093321*x^9 + 2002505540123*x^10 + ...
ILLUSTRATION OF DEFINITION.
(EX. 1) The table of coefficients of x^k in (1+x)^(n^2) / A(x) begins:
n=0: [1, -1,   -2,   -30,   -540,  -13380, -416910, -15634290, ...];
n=1: [1,  0,   -3,   -32,   -570,  -13920, -430290, -16051200, ...];
n=2: [1,  3,    0,   -40,   -675,  -15729, -473792, -17384400, ...];
n=3: [1,  8,   25,     0,   -840,  -19488, -559584, -19917600, ...];
n=4: [1, 15,  102,   378,      0,  -24192, -712590, -24272754, ...];
n=5: [1, 24,  273,  1920,   8460,       0, -883740, -31495200, ...];
n=6: [1, 35,  592,  6408,  48885,  252087,       0, -39049296, ...];
n=7: [1, 48, 1125, 17120, 189090, 1583040, 9392890,         0, ...]; ...
in which the main diagonal is all zeros after the initial term, illustrating that [x^n] (1+x)^(n^2) / A(x) = 0 for n > 0.
RELATED SEQUENCES.
(EX. 2) The secondary diagonal in the above table (EX. 1) that begins
[1, 3, 25, 378, 8460, 252087, 9392890, 420142350, ...]
yields A132617, column 1 of triangle A132615.
Related triangular matrix T = A132615 begins:
         1;
         1,       1;
         1,       1,      1;
         6,       3,      1,     1;
        80,      25,      5,     1,    1;
      1666,     378,     56,     7,    1,   1;
     47232,    8460,   1020,    99,    9,   1,  1;
   1694704,  252087,  26015,  2134,  154,  11,  1, 1;
  73552752, 9392890, 855478, 61919, 3848, 221, 13, 1, 1; ...
in which row n equals row (n-1) of T^(2*n-1) followed by '1' for n > 0.
(EX. 3) The next diagonal in the table (EX. 1) that begins:
[1, 8, 102, 1920, 48885, 1583040, 1583040, 62467314, ...]
yields the first column in the following matrix product.
Let TSL(m) denote the table T = A132615, with the diagonal of 1's truncated, as SHIFTED LEFT m times, so that
TSL(1) begins
  [   1];
  [   3,    1];
  [  25,    5,  1];
  [ 378,   56,  7, 1];
  [8460, 1020, 99, 9, 1]; ...
TSL(2) begins
  [    1];
  [    5,    1];
  [   56,    7,   1];
  [ 1020,   99,   9,  1];
  [26015, 2134, 154, 11, 1]; ...
etc.,
then the matrix product TSL(2)*TSL(1) begins
  [       1];
  [       8,       1];
  [     102,      12,      1];
  [    1920,     200,     16,     1];
  [   48885,    4540,    330,    20,   1];
  [ 1583040,  132810,   8816,   492,  24,  1];
  [62467314, 4790156, 293419, 15148, 686, 28, 1]; ...
in which the first column equals the secondary diagonal in the table of (EX. 1).
The subsequent diagonal in the table of (EX. 1) also equals the first column of matrix product TSL(3)*TSL(2)*TSL(1). This process can be continued to produce all the lower diagonals of the table of (EX. 1).

Paul D. Hanna, Table of n, a(n) for n = 0..300

Cf. A132617, A304190, A304192, A304193, A132615.

{a(n) = my(A=[1]); for(i=1,n, A=concat(A,0); m=#A; A[m] = Vec( (1+x +x*O(x^m))^((m-1)^2)/Ser(A) )[m] ); A[n+1]}
for(n=0,30, print1(a(n),", "))

A132617(n+1) = [x^n] (1+x)^((n+1)^2) / A(x) for n >= 0.

cp's OEIS Frontend

A132615 Triangle T, read by rows, where row n+1 of T = row n of T^(2n-1) with appended '1' for n>=0 with T(0,0)=1.

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A304193 G.f. A(x) satisfies: [x^n] (1+x)^((n+1)^2) / A(x) = 0 for n>0.

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A304191 G.f. A(x) satisfies: [x^n] (1+x)^(n^2) / A(x) = 0 for n > 0.

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