A302910 -id:A302910 - cp's OEIS frontend

A097591 Triangle read by rows: T(n,k) is the number of permutations of [n] with exactly k increasing runs of odd length.

1, 0, 1, 1, 0, 1, 0, 5, 0, 1, 6, 0, 17, 0, 1, 0, 70, 0, 49, 0, 1, 90, 0, 500, 0, 129, 0, 1, 0, 1890, 0, 2828, 0, 321, 0, 1, 2520, 0, 23100, 0, 13930, 0, 769, 0, 1, 0, 83160, 0, 215292, 0, 62634, 0, 1793, 0, 1, 113400, 0, 1549800, 0, 1697430, 0, 264072, 0, 4097, 0, 1

Offset: 0

Emeric Deutsch, Aug 29 2004

			Triangle starts:
     1;
     0,    1;
     1,    0,     1;
     0,    5,     0,    1;
     6,    0,    17,    0,     1;
     0,   70,     0,   49,     0,   1;
    90,    0,   500,    0,   129,   0,   1;
     0, 1890,     0, 2828,     0, 321,   0, 1;
  2520,    0, 23100,    0, 13930,   0, 769, 0, 1;
  ...
Row n has n+1 entries.
Example: T(3,1) = 5 because we have (123), 13(2), (2)13, 23(1) and (3)12 (the runs of odd length are shown between parentheses).

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

Bisections of columns k=0-1 give: A000680, A302910.
Row sums give A000142.
T(n+1,n-1) gives A000337.
T(4n,2n) gives A308962.
Cf. A096654, A097592.

G:=t^2/(1-t*x-(1-t^2)*exp(-t*x)): Gser:=simplify(series(G,x=0,12)): P[0]:=1: for n from 1 to 11 do P[n]:=sort(expand(n!*coeff(Gser,x^n))) od: seq(seq(coeff(t*P[n],t^k),k=1..n+1),n=0..11);
# second Maple program:
b:= proc(u, o, t) option remember; `if`(u+o=0, x^t, expand(
      add(b(u+j-1, o-j, irem(t+1, 2)), j=1..o)+
      add(b(u-j, o+j-1, 1)*x^t, j=1..u)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(b(n, 0, 1)):
seq(T(n), n=0..12);  # Alois P. Heinz, Nov 19 2013

b[u_, o_, t_] := b[u, o, t] = If[u+o == 0, x^t, Expand[Sum[b[u+j-1, o-j, Mod[t+1, 2]], {j, 1, o}] + Sum[b[u-j, o+j-1, 1]*x^t, {j, 1, u}]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 1, Exponent[p, x]}]][b[n, 0, 1]]; Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Feb 19 2015, after Alois P. Heinz *)

E.g.f.: t^2/[1-tx-(1-t^2)exp(-tx)].

Sum_{k=1..n} k * T(n,k) = A096654(n-1) for n > 0. - Alois P. Heinz, Jul 03 2019

A302909 Determinant of n X n matrix whose main diagonal consists of the first n 5-gonal numbers and all other elements are 1's.

Original entry on oeis.org

1, 4, 44, 924, 31416, 1570800, 108385200, 9863053200, 1144114171200, 164752440652800, 28831677114240000, 6025820516876160000, 1482351847151535360000, 423952628285339112960000, 139480414705876568163840000, 52305155514703713061440000000

Offset: 1

Muniru A Asiru, Apr 15 2018

nonn

			The 7 X 7 matrix (as below) has determinant 108385200.
  1  1  1  1  1  1  1
  1  5  1  1  1  1  1
  1  1 12  1  1  1  1
  1  1  1 22  1  1  1
  1  1  1  1 35  1  1
  1  1  1  1  1 51  1
  1  1  1  1  1  1 70

Muniru A Asiru, Table of n, a(n) for n = 1..100

Cf. A000326, A115067.

Cf. Determinant of n X n matrix whose main diagonal consists of the first n k-gonal numbers and all other elements are 1's: A000142 (k=2), A067550 (k=3), A010791 (k=4, with offset 1), this sequence (k=5), A302910 (k=6), A302911 (k=7), A302912 (k=8), A302913 (k=9), A302914 (k=10).

d:=(i,j)->`if`(i<>j,1,i*(3*i-1)/2):
seq(LinearAlgebra[Determinant](Matrix(n,d)),n=1..17);

Table[FullSimplify[Gamma[n] * Gamma[n + 5/3] * 3^(n + 1) / (5 * Gamma[2/3] * 2^n)], {n, 1, 15}] (* Vaclav Kotesovec, Apr 16 2018 *)
Module[{nn=20,pn5},pn5=PolygonalNumber[5,Range[nn]];Table[Det[DiagonalMatrix[Take[pn5,n]]/.(0->1)],{n,nn}]] (* Harvey P. Dale, Feb 07 2025 *)

a(n) = matdet(matrix(n, n, i, j, if (i!=j, 1, i*(3*i-1)/2))); \\ Michel Marcus, Apr 16 2018

first(n) = my(res = vector(n)); res[1] = 1; for(i = 1, n - 1, res[i + 1] = res[i] * i*(3*i + 5)/2); res \\ David A. Corneth, Apr 16 2018

From Vaclav Kotesovec, Apr 16 2018: (Start)
a(n) = Gamma(n) * Gamma(n + 5/3) * 3^(n + 1) / (5 * Gamma(2/3) * 2^n).
a(n) ~ Gamma(1/3) * 3^(n + 3/2) * n^(2*n + 2/3) / (5 * 2^n * exp(2*n)).
(End)
a(n + 1) = A115067(n + 1) * a(n) = a(n) * n*(3*n + 5)/2. - David A. Corneth, Apr 16 2018

A302911 Determinant of n X n matrix whose main diagonal consists of the first n 7-gonal numbers and all other elements are 1's.

Original entry on oeis.org

1, 6, 102, 3366, 181764, 14541120, 1614064320, 237267455040, 44606281547520, 10437869882119680, 2974792916404108800, 1014404384493801100800, 407790562566508042521600, 190845983281125763900108800, 102865984988526786742158643200

Offset: 1

Muniru A Asiru, Apr 15 2018

nonn

			The matrix begins:
  1   1   1   1   1   1   1 ...
  1   7   1   1   1   1   1 ...
  1   1  18   1   1   1   1 ...
  1   1   1  34   1   1   1 ...
  1   1   1   1  55   1   1 ...
  1   1   1   1   1  81   1 ...
  1   1   1   1   1   1 112 ...

Cf. A000566 (heptagonal numbers).

Cf. Determinant of n X n matrix whose main diagonal consists of the first n k-gonal numbers and all other elements are 1's: A000142 (k=2), A067550 (k=3), A010791 (k=4, with offset 1), A302909 (k=5), A302910 (k=6), this sequence (k=7), A302912 (k=8), A302913 (k=9), A302914 (k=10).

d:=(i,j)->`if`(i<>j,1,i*(5*i-3)/2):
seq(LinearAlgebra[Determinant](Matrix(n,d)),n=1..20);

nmax = 20; Table[Det[Table[If[i == j, i*(5*i - 3)/2, 1], {i, 1, k}, {j, 1, k}]], {k, 1, nmax}] (* Vaclav Kotesovec, Apr 16 2018 *)
Table[FullSimplify[5^(n + 1) * Gamma[n] * Gamma[n + 7/5] / (7 * Gamma[2/5] * 2^n)], {n, 1, 15}] (* Vaclav Kotesovec, Apr 16 2018 *)

a(n) = matdet(matrix(n, n, i, j, if (i!=j, 1, i*(5*i-3)/2))); \\ Michel Marcus, Apr 16 2018

From Vaclav Kotesovec, Apr 16 2018: (Start)
a(n) = 5^(n + 1) * Gamma(n) * Gamma(n + 7/5) / (7 * Gamma(2/5) * 2^n).
a(n) ~ Pi * 5^(n+1) * n^(2*n + 2/5) / (7 * Gamma(2/5) * 2^(n-1) * exp(2*n)).
a(n+1) = a(n) * n*(5*n + 7)/2.
(End)

A302912 Determinant of n X n matrix whose main diagonal consists of the first n 8-gonal numbers and all other elements are 1's.

Original entry on oeis.org

1, 7, 140, 5460, 349440, 33196800, 4381977600, 766846080000, 171773521920000, 47924812615680000, 16294436289331200000, 6631835569757798400000, 3183281073483743232000000, 1779454120077412466688000000, 1145968453329853628547072000000

Offset: 1

Muniru A Asiru, Apr 15 2018

nonn

			The matrix begins:
  1   1   1   1   1   1   1 ...
  1   8   1   1   1   1   1 ...
  1   1  21   1   1   1   1 ...
  1   1   1  40   1   1   1 ...
  1   1   1   1  65   1   1 ...
  1   1   1   1   1  96   1 ...
  1   1   1   1   1   1 133 ...

Cf. A000567 (octagonal numbers).

Cf. Determinant of n X n matrix whose main diagonal consists of the first n k-gonal numbers and all other elements are 1's: A000142 (k=2), A067550 (k=3), A010791 (k=4, with offset 1), A302909 (k=5), A302910 (k=6), A302911 (k=7), this sequence (k=8), A302913 (k=9), A302914 (k=10).

d:=(i,j)->`if`(i<>j,1,i*(3*i-2)):
seq(LinearAlgebra[Determinant](Matrix(n,d)),n=1..16);

nmax = 20; Table[Det[Table[If[i == j, i*(3*i - 2), 1], {i, 1, k}, {j, 1, k}]], {k, 1, nmax}] (* Vaclav Kotesovec, Apr 16 2018 *)
Table[FullSimplify[3^(n+1) * Gamma[n] * Gamma[n + 4/3] / (4*Gamma[1/3])], {n, 1, 15}] (* Vaclav Kotesovec, Apr 16 2018 *)RecurrenceTable[{a[n+1] == a[n] * n * (3*n + 4), a[1] == 1}, a, {n, 1, 20}] (* Vaclav Kotesovec, Apr 16 2018 *)

a(n) = matdet(matrix(n, n, i, j, if (i!=j, 1, i*(3*i-2)))); \\ Michel Marcus, Apr 16 2018

From Vaclav Kotesovec, Apr 16 2018: (Start)
a(n) = 3^(n+1) * Gamma(n) * Gamma(n + 4/3) / (4*Gamma(1/3)).
a(n) ~ Pi * 3^(n+1) * n^(2*n + 1/3) / (2 * Gamma(1/3) * exp(2*n)).
a(n+1) = a(n) * n*(3*n + 4).
(End)

A302913 Determinant of n X n matrix whose main diagonal consists of the first n 9-gonal numbers and all other elements are 1's.

Original entry on oeis.org

1, 8, 184, 8280, 612720, 67399200, 10312077600, 2093351752800, 544271455728000, 176343951655872000, 69655860904069440000, 32947222207624845120000, 18384549991854663576960000, 11949957494705531325024000000, 8950518163534442962442976000000

Offset: 1

Muniru A Asiru, Apr 15 2018

nonn

			The matrix begins:
1   1   1   1   1   1   1 ...
1   9   1   1   1   1   1 ...
1   1  24   1   1   1   1 ...
1   1   1  46   1   1   1 ...
1   1   1   1  75   1   1 ...
1   1   1   1   1 111   1 ...
1   1   1   1   1   1 154 ...

Cf. A001106 (nonagonal numbers).

Cf. Determinant of n X n matrix whose main diagonal consists of the first n k-gonal numbers and all other elements are 1's: A000142 (k=2), A067550 (k=3), A010791 (k=4, with offset 1), A302909 (k=5), A302910 (k=6), A302911 (k=7), A302912 (k=8), this sequence (k=9), A302914 (k=10).

d:=(i,j)->`if`(i<>j,1,i*(7*i-5)/2):
seq(LinearAlgebra[Determinant](Matrix(n,d)),n=1..16);

nmax = 20; Table[Det[Table[If[i == j, i*(7*i-5)/2, 1], {i, 1, k}, {j, 1, k}]], {k, 1, nmax}] (* Vaclav Kotesovec, Apr 16 2018 *)
RecurrenceTable[{a[n+1] == a[n] * n*(7*n + 9)/2, a[1] == 1}, a, {n, 1, 20}] (* Vaclav Kotesovec, Apr 16 2018 *)
Table[FullSimplify[7^(n + 1) * Gamma[n] * Gamma[n + 9/7] / (9*Gamma[2/7]*2^n)], {n, 1, 15}] (* Vaclav Kotesovec, Apr 16 2018 *)

a(n) = matdet(matrix(n, n, i, j, if (i!=j, 1, i*(7*i-5)/2))); \\ Michel Marcus, Apr 16 2018

From Vaclav Kotesovec, Apr 16 2018: (Start)
a(n) = 7^(n+1) * Gamma(n) * Gamma(n + 9/7) / (9 * Gamma(2/7) * 2^n).
a(n) ~ Pi * 7^(n+1) * n^(2*n + 2/7) / (9 * Gamma(2/7) * 2^(n-1) * exp(2*n)).
a(n+1) = a(n) * n*(7*n + 9)/2.
(End)

A302914 Determinant of n X n matrix whose main diagonal consists of the first n 10-gonal numbers and all other elements are 1's.

Original entry on oeis.org

1, 9, 234, 11934, 1002456, 125307000, 21803418000, 5036589558000, 1490830509168000, 550116457882992000, 247552406047346400000, 133430746859519709600000, 84861955002654535305600000, 62882708656967010661449600000, 53701833193049827104877958400000

Offset: 1

Muniru A Asiru, Apr 15 2018

nonn

From Vaclav Kotesovec, Apr 16 2018: (Start)
In general, for k > 2, these determinants for k-gonal numbers satisfies:
a(n,k) = ((k-2)/2)^(n-1) * Gamma(n) * Gamma(n + k/(k-2)) / Gamma(1 + k/(k-2)).
a(n,k) ~ 4*Pi * (k/2 - 1)^n * n^(2*n + 2/(k-2)) / (k * Gamma(k/(k-2)) * exp(2*n)).
a(n+1,k) = a(n,k) * n*((k-2)*n + k)/2.
(End)

			The matrix begins:
  1   1   1   1   1   1   1 ...
  1  10   1   1   1   1   1 ...
  1   1  27   1   1   1   1 ...
  1   1   1  52   1   1   1 ...
  1   1   1   1  85   1   1 ...
  1   1   1   1   1 126   1 ...
  1   1   1   1   1   1 175 ...

Cf. A001107.
Cf. Determinant of n X n matrix whose main diagonal consists of the first n k-gonal numbers and all other elements are 1's: A000142 (k=2), A067550 (k=3), A010791 (k=4, with offset 1), A302909 (k=5), A302910 (k=6), A302911 (k=7), A302912 (k=8), A302913 (k=9), this sequence (k=10).
Cf. A007840 (permanent instead of determinant, for k=2).

d:=(i,j)->`if`(i<>j,1,i*(4*i-3)):
seq(LinearAlgebra[Determinant](Matrix(n,d)),n=1..16);

nmax = 20; Table[Det[Table[If[i == j, i*(4*i-3), 1], {i, 1, k}, {j, 1, k}]], {k, 1, nmax}] (* Vaclav Kotesovec, Apr 16 2018 *)
RecurrenceTable[{a[n+1] == a[n] * n*(4*n + 5), a[1] == 1}, a, {n, 1, 20}] (* Vaclav Kotesovec, Apr 16 2018 *)
Table[FullSimplify[4^(n+1) * Gamma[n] * Gamma[n + 5/4] / (5*Gamma[1/4])], {n, 1, 15}] (* Vaclav Kotesovec, Apr 16 2018 *)

a(n) = matdet(matrix(n, n, i, j, if (i!=j, 1, i*(4*i-3)))); \\ Michel Marcus, Apr 16 2018

From Vaclav Kotesovec, Apr 16 2018: (Start)
a(n) = 4^(n+1) * Gamma(n) * Gamma(n + 5/4) / (5*Gamma(1/4)).
a(n) ~ Pi * 2^(2*n + 3) * n^(2*n + 1/4) / (5 * Gamma(1/4) * exp(2*n)).
a(n+1) = a(n) * n*(4*n + 5).
(End)

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A097591 Triangle read by rows: T(n,k) is the number of permutations of [n] with exactly k increasing runs of odd length.

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A302909 Determinant of n X n matrix whose main diagonal consists of the first n 5-gonal numbers and all other elements are 1's.

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A302911 Determinant of n X n matrix whose main diagonal consists of the first n 7-gonal numbers and all other elements are 1's.

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A302912 Determinant of n X n matrix whose main diagonal consists of the first n 8-gonal numbers and all other elements are 1's.

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Maple

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A302913 Determinant of n X n matrix whose main diagonal consists of the first n 9-gonal numbers and all other elements are 1's.

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A302914 Determinant of n X n matrix whose main diagonal consists of the first n 10-gonal numbers and all other elements are 1's.

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