A129186 -id:A129186 - cp's OEIS frontend

A194705 Triangle read by rows: T(k,m) = number of occurrences of k in the last section of the set of partitions of (5 + m).

7, 4, 3, 2, 2, 3, 1, 1, 3, 2, 0, 1, 1, 2, 3, 1, 0, 1, 1, 2, 2, 0, 1, 0, 1, 1, 2, 2, 0, 0, 1, 0, 1, 1, 2, 2, 0, 0, 0, 1, 0, 1, 1, 2, 2, 0, 0, 0, 0, 1, 0, 1, 1, 2, 2, 0, 0, 0, 0, 0, 1, 0, 1, 1, 2, 2, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 2, 2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 2, 2

Offset: 1

Omar E. Pol, Feb 05 2012

Sub-triangle of A182703 and also of A194812. Note that the sum of every row is also the number of partitions of 5. For further information see A182703 and A135010.

			Triangle begins:
  7,
  4, 3,
  2, 2, 3,
  1, 1, 3, 2,
  0, 1, 1, 2, 3,
  1, 0, 1, 1, 2, 2,
  0, 1, 0, 1, 1, 2, 2,
  ...
For k = 1 and m = 1: T(1,1) = 7 because there are seven parts of size 1 in the last section of the set of partitions of 6, since 5 + m = 6, so a(1) = 7.
For k = 2 and m = 1: T(2,1) = 4 because there are four parts of size 2 in the last section of the set of partitions of 6, since 5 + m = 6, so a(2) = 4.

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)

Always the sum of row k = p(5) = A000041(5) = 7.
The first (0-10) members of this family of triangles are A023531, A129186, A194702-A194704, this sequence, A194706-A194710.
Cf. A135010, A138121, A194812.

P(n)={my(M=matrix(n,n), d=5); M[1,1]=numbpart(d); for(m=1, n, forpart(p=m+d, for(k=1, #p, my(t=p[k]); if(t<=n && m<=t, M[t, m]++)), [2, m+d])); M}
{ my(T=P(10)); for(n=1, #T, print(T[n, 1..n])) } \\ Andrew Howroyd, Feb 19 2020

T(k,m) = A182703(5+m,k), with T(k,m) = 0 if k > 5+m.

T(k,m) = A194812(5+m,k).

Terms a(29) and beyond from Andrew Howroyd, Feb 19 2020

A194706 Triangle read by rows: T(k,m) = number of occurrences of k in the last section of the set of partitions of (6 + m).

Original entry on oeis.org

11, 3, 8, 2, 3, 6, 1, 3, 2, 5, 1, 1, 2, 3, 4, 0, 1, 1, 2, 2, 5, 1, 0, 1, 1, 2, 2, 4, 0, 1, 0, 1, 1, 2, 2, 4, 0, 0, 1, 0, 1, 1, 2, 2, 4, 0, 0, 0, 1, 0, 1, 1, 2, 2, 4, 0, 0, 0, 0, 1, 0, 1, 1, 2, 2, 4, 0, 0, 0, 0, 0, 1, 0, 1, 1, 2, 2, 4, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 2, 2, 4

Offset: 1

Omar E. Pol, Feb 05 2012

Sub-triangle of A182703 and also of A194812. Note that the sum of every row is also the number of partitions of 6. For further information see A182703 and A135010.

			Triangle begins:
  11,
   3, 8,
   2, 3, 6,
   1, 3, 2, 5,
   1, 1, 2, 3, 4,
   0, 1, 1, 2, 2, 5,
  ...
For k = 1 and m = 1: T(1,1) = 11 because there are 11 parts of size 1 in the last section of the set of partitions of 7, since 6 + m = 7, so a(1) = 11.
For k = 2 and m = 1: T(2,1) = 3 because there are three parts of size 2 in the last section of the set of partitions of 7, since 6 + m = 7, so a(2) = 3.

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)

Always the sum of row k = p(6) = A000041(6) = 11.
The first (0-10) members of this family of triangles are A023531, A129186, A194702-A194705, this sequence, A194707-A194710.
Cf. A135010, A138121, A194812.

P(n)={my(M=matrix(n,n), d=6); M[1,1]=numbpart(d); for(m=1, n, forpart(p=m+d, for(k=1, #p, my(t=p[k]); if(t<=n && m<=t, M[t, m]++)), [2, m+d])); M}
{ my(T=P(10)); for(n=1, #T, print(T[n, 1..n])) } \\ Andrew Howroyd, Feb 19 2020

T(k,m) = A182703(6+m,k), with T(k,m) = 0 if k > 6+m.

T(k,m) = A194812(6+m,k).

Terms a(22) and beyond from Andrew Howroyd, Feb 19 2020

A194707 Triangle read by rows: T(k,m) = number of occurrences of k in the last section of the set of partitions of (7 + m).

Original entry on oeis.org

15, 8, 7, 3, 6, 6, 3, 2, 5, 5, 1, 2, 3, 4, 5, 1, 1, 2, 2, 5, 4, 0, 1, 1, 2, 2, 4, 5, 1, 0, 1, 1, 2, 2, 4, 4, 0, 1, 0, 1, 1, 2, 2, 4, 4, 0, 0, 1, 0, 1, 1, 2, 2, 4, 4, 0, 0, 0, 1, 0, 1, 1, 2, 2, 4, 4, 0, 0, 0, 0, 1, 0, 1, 1, 2, 2, 4, 4, 0, 0, 0, 0, 0, 1, 0, 1, 1, 2, 2, 4, 4

Offset: 1

Omar E. Pol, Feb 05 2012

Sub-triangle of A182703 and also of A194812. Note that the sum of every row is also the number of partitions of 7. For further information see A182703 and A135010.

			Triangle begins:
  15,
   8, 7,
   3, 6, 6,
   3, 2, 5, 5,
  ...
For k = 1 and m = 1: T(1,1) = 15 because there are 15 parts of size 1 in the last section of the set of partitions of 8, since 7 + m = 8, so a(1) = 15.
For k = 2 and m = 1: T(2,1) = 8 because there are eight parts of size 2 in the last section of the set of partitions of 8, since 7 + m = 8, so a(2) = 8.

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)

Always the sum of row k = p(7) = A000041(7) = 15.
The first (0-10) members of this family of triangles are A023531, A129186, A194702-A194706, this sequence, A194708-A194710.
Cf. A135010, A138121, A194812.

P(n)={my(M=matrix(n,n), d=7); M[1,1]=numbpart(d); for(m=1, n, forpart(p=m+d, for(k=1, #p, my(t=p[k]); if(t<=n && m<=t, M[t, m]++)), [2, m+d])); M}
{ my(T=P(10)); for(n=1, #T, print(T[n, 1..n])) } \\ Andrew Howroyd, Feb 19 2020

T(k,m) = A182703(7+m,k), with T(k,m) = 0 if k > 7+m.

T(k,m) = A194812(7+m,k).

Terms a(11) and beyond from Andrew Howroyd, Feb 19 2020

A194708 Triangle read by rows: T(k,m) = number of occurrences of k in the last section of the set of partitions of (8 + m).

Original entry on oeis.org

22, 7, 15, 6, 6, 10, 2, 5, 5, 10, 2, 3, 4, 5, 8, 1, 2, 2, 5, 4, 8, 1, 1, 2, 2, 4, 5, 7, 0, 1, 1, 2, 2, 4, 4, 8, 1, 0, 1, 1, 2, 2, 4, 4, 7, 0, 1, 0, 1, 1, 2, 2, 4, 4, 7, 0, 0, 1, 0, 1, 1, 2, 2, 4, 4, 7, 0, 0, 0, 1, 0, 1, 1, 2, 2, 4, 4, 7, 0, 0, 0, 0, 1, 0, 1, 1, 2, 2, 4, 4, 7

Offset: 1

Omar E. Pol, Feb 05 2012

Sub-triangle of A182703 and also of A194812. Note that the sum of every row is also the number of partitions of 8. For further information see A182703 and A135010.

			Triangle begins:
  22,
   7, 15,
   6,  6, 10,
   2,  5,  5, 10,
   2,  3,  4,  5,  8,
   ...
For k = 1 and m = 1: T(1,1) = 22 because there are 22 parts of size 1 in the last section of the set of partitions of 9, since 8 + m = 9, so a(1) = 22.
For k = 2 and m = 1: T(2,1) = 7 because there are seven parts of size 2 in the last section of the set of partitions of 9, since 8 + m = 9, so a(2) = 7.

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)

Always the sum of row k = p(8) = A000041(8) = 22.
The first (0-10) members of this family of triangles are A023531, A129186, A194702-A194707, this sequence, A194709, A194710.
Cf. A135010, A138121, A194812.

P(n)={my(M=matrix(n,n), d=8); M[1,1]=numbpart(d); for(m=1, n, forpart(p=m+d, for(k=1, #p, my(t=p[k]); if(t<=n && m<=t, M[t, m]++)), [2, m+d])); M}
{ my(T=P(10)); for(n=1, #T, print(T[n, 1..n])) } \\ Andrew Howroyd, Feb 19 2020

T(k,m) = A182703(8+m,k), with T(k,m) = 0 if k > 8+m.

T(k,m) = A194812(8+m,k).

Terms a(11) and beyond from Andrew Howroyd, Feb 19 2020

A194709 Triangle read by rows: T(k,m) = number of occurrences of k in the last section of the set of partitions of (9 + m).

Original entry on oeis.org

30, 15, 15, 6, 10, 14, 5, 5, 10, 10, 3, 4, 5, 8, 10, 2, 2, 5, 4, 8, 9, 1, 2, 2, 4, 5, 7, 9, 1, 1, 2, 2, 4, 4, 8, 8, 0, 1, 1, 2, 2, 4, 4, 7, 9, 1, 0, 1, 1, 2, 2, 4, 4, 7, 8, 0, 1, 0, 1, 1, 2, 2, 4, 4, 7, 8, 0, 0, 1, 0, 1, 1, 2, 2, 4, 4, 7, 8, 0, 0, 0, 1, 0, 1, 1, 2, 2, 4, 4, 7, 8

Offset: 1

Omar E. Pol, Feb 05 2012

Sub-triangle of A182703 and also of A194812. Note that the sum of every row is also the number of partitions of 9. For further information see A182703 and A135010.

			Triangle begins:
  30;
  15, 15;
   6, 10, 14;
   5,  5, 10, 10;
   3,  4,  5,  8, 10;
   2,  2,  5,  4,  8, 9;
  ...
For k = 1 and  m = 1; T(1,1) = 30 because there are 30 parts of size 1 in the last section of the set of partitions of 10, since 9 + m = 10, so a(1) = 30. For k = 2 and m = 1; T(2,1) = 15 because there are 15 parts of size 2 in the last section of the set of partitions of 10, since 9 + m = 10, so a(2) = 15.

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)

Always the sum of row k = p(9) = A000041(n) = 30.
The first (0-10) members of this family of triangles are A023531, A129186, A194702-A194708, this sequence, A194710.
Cf. A135010, A138121, A194812.

P(n)={my(M=matrix(n,n), d=9); M[1,1]=numbpart(d); for(m=1, n, forpart(p=m+d, for(k=1, #p, my(t=p[k]); if(t<=n && m<=t, M[t, m]++)), [2, m+d])); M}
{ my(T=P(10)); for(n=1, #T, print(T[n, 1..n])) } \\ Andrew Howroyd, Feb 19 2020

T(k,m) = A182703(9+m,k), with T(k,m) = 0 if k > 9+m.

T(k,m) = A194812(9+m,k).

Terms a(7) and beyond from Andrew Howroyd, Feb 19 2020

A292605 Triangle read by rows, coefficients of generalized Eulerian polynomials F_{3;n}(x).

Original entry on oeis.org

1, 1, 0, 19, 1, 0, 1513, 166, 1, 0, 315523, 52715, 1361, 1, 0, 136085041, 30543236, 1528806, 10916, 1, 0, 105261234643, 29664031413, 2257312622, 42421946, 87375, 1, 0, 132705221399353, 45011574747714, 4637635381695, 153778143100, 1156669095, 699042, 1, 0

Offset: 0

Peter Luschny, Sep 20 2017

See the comments in A292604.

			Triangle starts:
[n\k][       0         1         2       3  4  5]
--------------------------------------------------
[0][         1]
[1][         1,        0]
[2][        19,        1,        0]
[3][      1513,      166,        1,     0]
[4][    315523,    52715,     1361,     1,  0]
[5][ 136085041, 30543236,  1528806, 10916,  1, 0]

F_{0} = A129186, F_{1} = A173018, F_{2} = A292604, F_{3} is this triangle, F_{4} = A292606.
First column: A002115. Row sums: A014606. Alternating row sums: A292609.
Cf. A181985, A278073.

Coeffs := f -> PolynomialTools:-CoefficientList(expand(f),x):
A292605_row := proc(n) if n = 0 then return [1] fi;
add(A278073(n, k)*(x-1)^(n-k), k=0..n); [op(Coeffs(%)), 0] end:
for n from 0 to 6 do A292605_row(n) od;

# uses[A278073_row from A278073]
def A292605_row(n):
    if n == 0: return [1]
    L = A278073_row(n)
    S = sum(L[k]*(x-1)^(n-k) for k in (0..n))
    return expand(S).list() + [0]
for n in (0..5): print(A292605_row(n))

F_{3; n}(x) = Sum_{k=0..n} A278073(n, k)*(x-1)^(n-k) for n>0 and F_{3; 0}(x) = 1.

A292606 Triangle read by rows, coefficients of generalized Eulerian polynomials F_{4;n}(x).

Original entry on oeis.org

1, 1, 0, 69, 1, 0, 33661, 988, 1, 0, 60376809, 2669683, 16507, 1, 0, 288294050521, 17033188586, 212734266, 261626, 1, 0, 3019098162602349, 223257353561605, 4297382231090, 17634518610, 4196345, 1, 0

Offset: 0

Peter Luschny, Sep 26 2017

See the comments in A292604.

			Triangle starts:
[n\k][          0            1          2       3   4   5]
--------------------------------------------------
[0] [           1]
[1] [           1,           0]
[2] [          69,           1,         0]
[3] [       33661,         988,         1,      0]
[4] [    60376809,     2669683,     16507,      1,  0]
[5] [288294050521, 17033188586, 212734266, 261626,  1,  0]

F_{0} = A129186, F_{1} = A173018, F_{2} = A292604, F_{3} = A292605, F_{4} is this triangle.
First column: A211212. Row sums: A014608. Alternating row sums: A292607.
Cf. A181985.

Coeffs := f -> PolynomialTools:-CoefficientList(expand(f), x):
A292606_row := proc(n) if n = 0 then return [1] fi;
add(A278074(n, k)*(x-1)^(n-k), k=0..n); [op(Coeffs(%)), 0] end:
for n from 0 to 6 do A292606_row(n) od;

# uses[A278074_row from A278074]
def A292606_row(n):
    if n == 0: return [1]
    L = A278074_row(n)
    S = sum(L[k]*(x-1)^(n-k) for k in (0..n))
    return expand(S).list() + [0]
for n in (0..5): print(A292606_row(n))

F_{4; n}(x) = Sum_{k=0..n} A278074(n, k)*(x-1)^(n-k) for n>0 and F_{4; 0}(x) = 1.

A183190 Triangle T(n,k), read by rows, given by (1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 1, 0, 2, 1, 0, 4, 4, 1, 0, 8, 12, 6, 1, 0, 16, 32, 24, 8, 1, 0, 32, 80, 80, 40, 10, 1, 0, 64, 192, 240, 160, 60, 12, 1, 0, 128, 448, 672, 560, 280, 84, 14, 1, 0, 256, 1024, 1792, 1792, 1120, 448, 112, 16, 1, 0, 512, 2304, 4608, 5376, 4032, 2016, 672, 144, 18, 1, 0

Offset: 0

Philippe Deléham, Dec 14 2011

A071919*A007318 as infinite lower triangular matrices.
A129186*A038207 as infinite lower triangular matrices.
From Paul Curtz, Nov 12 2019: (Start)
If a new main diagonal of 0's is added to the triangle, then for this variant the following propositions hold:
   The first column is A166444.
   The second column is A139756.
   The antidiagonal sums are A000129 (Pell numbers).
   The row sums are (-1)^n*A141413.
   The signed row sums are 0 followed by 1's, autosequence companion to A054977.
(End)

			Triangle begins:
   1;
   1,  0;
   2,  1,  0;
   4,  4,  1,  0;
   8, 12,  6,  1,  0;
  16, 32, 24,  8,  1, 0;
  32, 80, 80, 40, 10, 1, 0;
  ...

Essentially the same as A038207, A062715, A065109.
Cf. A001787, A001788, A139756, A000129 (antidiagonals sums).
Cf. A141413, A166444, A054977.

T:= proc(n, k) option remember; `if`(k<0 or k>n, 0,
      `if`(n<2, 1-k, 2*T(n-1, k) +T(n-1, k-1)))
    end:
seq(seq(T(n,k), k=0..n), n=0..12);  # Alois P. Heinz, Nov 08 2019

T[n_, k_] /; 0 <= k <= n := T[n, k] = 2 T[n-1, k] + T[n-1, k-1];
T[0, 0] = T[1, 0] = 1; T[1, 1] = 0; T[, ] = 0;
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 08 2019 *)

T(n,k) = 2*T(n-1,k) + T(n-1,k-1) with T(0,0)=T(1,0)=1 and T(1,1)=0 .
G.f.: (1-(1+y)*x)/(1-(2+y)*x).
Sum_{k, 0<=k<=n} T(n,k)*x^k = A019590(n+1), A000012(n), A011782(n), A133494(n) for x = -2, -1, 0, 1 respectively.
Sum_{k, 0<=k<=n} T(n,k)*x^(n-k) = A000007(n), A133494(n), A020699(n) for x = 0, 1, 2 respectively.
T(2n,n) = A069720(n).

A194703 Triangle read by rows: T(k,m) = number of occurrences of k in the last section of the set of partitions of (3 + m).

Original entry on oeis.org

3, 2, 1, 0, 1, 2, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 3, 2, 1, 0, 1, 2, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0

Offset: 1

Omar E. Pol, Feb 05 2012

Sub-triangle of A182703 and also of A194812. Note that the sum of every row is also the number of partitions of 3. For further information see A182703 and A135010.

			Triangle begins:
3,
2, 1,
0, 1, 2,
1, 0, 1, 1,
0, 1, 0, 1, 1,
0, 0, 1, 0, 1, 1,
0, 0, 0, 1, 0, 1, 1,
0, 0, 0, 0, 1, 0, 1, 1,
0, 0, 0, 0, 0, 1, 0, 1, 1,
0, 0, 0, 0, 0, 0, 1, 0, 1, 1,
...
For k = 1 and m = 1, T(1,1) = 3 because there are three parts of size 1 in the last section of the set of partitions of 4, since 3 + m = 4, so a(1) = 3.
For k = 2 and m = 1, T(2,1) = 2 because there are two parts of size 2 in the last section of the set of partitions of 4, since 3 + m = 4, so a(2) = 2.

Always the sum of row k = p(3) = A000041(3) = 3.
The first (0-10) members of this family of triangles are A023531, A129186, A194702, this sequence, A194704-A194710.
Cf. A135010, A138121, A182712-A182714, A194812.

T(k,m) = A182703(3+m,k), with T(k,m) = 0 if k > 3+m.

T(k,m) = A194812(3+m,k).

cp's OEIS Frontend

A194705 Triangle read by rows: T(k,m) = number of occurrences of k in the last section of the set of partitions of (5 + m).

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A194706 Triangle read by rows: T(k,m) = number of occurrences of k in the last section of the set of partitions of (6 + m).

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A194707 Triangle read by rows: T(k,m) = number of occurrences of k in the last section of the set of partitions of (7 + m).

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A194708 Triangle read by rows: T(k,m) = number of occurrences of k in the last section of the set of partitions of (8 + m).

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A194709 Triangle read by rows: T(k,m) = number of occurrences of k in the last section of the set of partitions of (9 + m).

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A292605 Triangle read by rows, coefficients of generalized Eulerian polynomials F_{3;n}(x).

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A292606 Triangle read by rows, coefficients of generalized Eulerian polynomials F_{4;n}(x).

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