A094002 -id:A094002 - cp's OEIS frontend

A157207 Triangle T(n, k, m) = (m(n-k) + 1)T(n-1, k-1, m) + (mk + 1)T(n-1, k, m) + mf(n,k)T(n-2, k-1, m) with T(n, 0, m) = T(n, n, m) = 1, f(n, k) = k if k <= floor(n/2) otherwise n-k, and m = 1, read by rows.

1, 1, 1, 1, 5, 1, 1, 14, 14, 1, 1, 33, 94, 33, 1, 1, 72, 442, 442, 72, 1, 1, 151, 1752, 3818, 1752, 151, 1, 1, 310, 6306, 25358, 25358, 6306, 310, 1, 1, 629, 21390, 144524, 268852, 144524, 21390, 629, 1, 1, 1268, 69822, 746744, 2312836, 2312836, 746744, 69822, 1268, 1

Offset: 0

Roger L. Bagula, Feb 25 2009

			Triangle begins as:
  1;
  1,    1;
  1,    5,     1;
  1,   14,    14,      1;
  1,   33,    94,     33,       1;
  1,   72,   442,    442,      72,       1;
  1,  151,  1752,   3818,    1752,     151,      1;
  1,  310,  6306,  25358,   25358,    6306,    310,     1;
  1,  629, 21390, 144524,  268852,  144524,  21390,   629,    1;
  1, 1268, 69822, 746744, 2312836, 2312836, 746744, 69822, 1268, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A007318 (m=0), this sequence (m=1), A157208 (m=2), A157209 (m=3).
Cf. A157147, A157148, A157149, A157150, A157151, A157152, A157153, A157154, A157155, A157156, A157210, A157211, A157212, A157268, A157272, A157273, A157274, A157275.
Cf. A094002.

f[n_,k_]:= If[k<=Floor[n/2], k, n-k];
T[n_, k_, m_]:= T[n, k, m]= If[k==0 || k==n, 1, (m*(n-k)+1)*T[n-1,k-1,m] + (m*k+1)*T[n-1,k,m] + m*f[n,k]*T[n-2,k-1,m]];
Table[T[n,k,1], {n,0,10}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Jan 10 2022 *)

def f(n,k): return k if (k <= n//2) else n-k
@CachedFunction
def T(n,k,m):  # A157207
    if (k==0 or k==n): return 1
    else: return (m*(n-k) +1)*T(n-1,k-1,m) + (m*k+1)*T(n-1,k,m) + m*f(n,k)*T(n-2,k-1,m)
flatten([[T(n,k,1) for k in (0..n)] for n in (0..20)]) # G. C. Greubel, Jan 10 2022

T(n, k, m) = (m*(n-k) + 1)*T(n-1, k-1, m) + (m*k + 1)*T(n-1, k, m) + m*f(n,k)*T(n-2, k-1, m) with T(n, 0, m) = T(n, n, m) = 1, f(n, k) = k if k <= floor(n/2) otherwise n-k, and m = 1.
T(n, n-k, m) = T(n, k, m).
T(n, 1, 1) = A094002(n-1). - G. C. Greubel, Jan 10 2022

Edited by G. C. Greubel, Jan 10 2022

A274835 Number A(n,k) of set partitions of [n] such that the difference between each element and its block index is a multiple of k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 5, 1, 1, 1, 1, 2, 15, 1, 1, 1, 1, 1, 3, 52, 1, 1, 1, 1, 1, 2, 7, 203, 1, 1, 1, 1, 1, 1, 3, 14, 877, 1, 1, 1, 1, 1, 1, 2, 4, 39, 4140, 1, 1, 1, 1, 1, 1, 1, 3, 9, 95, 21147, 1, 1, 1, 1, 1, 1, 1, 2, 4, 18, 304, 115975, 1

Offset: 0

Alois P. Heinz, Jul 08 2016

			A(3,0) = 1: 1|2|3.
A(3,1) = 5: 123, 12|3, 13|2, 1|23, 1|2|3.
A(5,2) = 7: 135|24, 13|24|5, 15|24|3, 1|24|35, 15|2|3|4, 1|2|35|4, 1|2|3|4|5.
A(7,3) = 9: 147|25|36, 14|25|36|7, 17|25|36|4, 1|25|36|47, 17|2|36|4|5, 1|2|36|47|5, 17|2|3|4|5|6, 1|2|3|47|5|6, 1|2|3|4|5|6|7.
Square array A(n,k) begins:
  1,      1,   1,  1,  1, 1, 1, 1, 1, 1, 1, ...
  1,      1,   1,  1,  1, 1, 1, 1, 1, 1, 1, ...
  1,      2,   1,  1,  1, 1, 1, 1, 1, 1, 1, ...
  1,      5,   2,  1,  1, 1, 1, 1, 1, 1, 1, ...
  1,     15,   3,  2,  1, 1, 1, 1, 1, 1, 1, ...
  1,     52,   7,  3,  2, 1, 1, 1, 1, 1, 1, ...
  1,    203,  14,  4,  3, 2, 1, 1, 1, 1, 1, ...
  1,    877,  39,  9,  4, 3, 2, 1, 1, 1, 1, ...
  1,   4140,  95, 18,  5, 4, 3, 2, 1, 1, 1, ...
  1,  21147, 304, 33, 11, 5, 4, 3, 2, 1, 1, ...
  1, 115975, 865, 89, 22, 6, 5, 4, 3, 2, 1, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
Wikipedia, Partition of a set

Columns k=0-10 give: A000012, A000110, A274538, A274836, A274837, A274838, A274839, A274840, A274841, A274842, A274843.
Main diagonal gives A000012.
A(n,ceiling(n/2)) gives A008619.
A(3n,n) gives A094002.

b:= proc(n, k, m, t) option remember; `if`(n=0, 1,
     add(`if`(irem(j-t, k)=0, b(n-1, k, max(m, j),
              irem(t+1, k)), 0), j=1..m+1))
    end:
A:= (n, k)-> `if`(k=0, 1, b(n, k, 0, 1)):
seq(seq(A(n, d-n), n=0..d), d=0..14);

b[n_, k_, m_, t_] := b[n, k, m, t] = If[n==0, 1, Sum[If[Mod[j-t, k]==0, b[n-1, k, Max[m, j], Mod[t+1, k]], 0], {j, 1, m+1}]]; A[n_, k_]:= If[k==0, 1, b[n, k, 0, 1]]; Table[A[n, d-n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-François Alcover, Dec 18 2016, after Alois P. Heinz *)

A144438 Triangle T(n,k) by rows: T(n, k) = (n-k+1)T(n-1, k-1) + kT(n-1, k) + T(n-2, k-1) with T(n, 1) = T(n, n) = 1.

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 1, 14, 14, 1, 1, 33, 89, 33, 1, 1, 72, 413, 413, 72, 1, 1, 151, 1632, 3393, 1632, 151, 1, 1, 310, 5874, 22145, 22145, 5874, 310, 1, 1, 629, 19943, 125456, 224843, 125456, 19943, 629, 1, 1, 1268, 65171, 647299, 1899096, 1899096, 647299, 65171, 1268, 1

Offset: 1

Roger L. Bagula and Gary W. Adamson, Oct 05 2008

			Triangle begins as:
  1;
  1,    1;
  1,    5,     1;
  1,   14,    14,      1;
  1,   33,    89,     33,       1;
  1,   72,   413,    413,      72,       1;
  1,  151,  1632,   3393,    1632,     151,      1;
  1,  310,  5874,  22145,   22145,    5874,    310,     1;
  1,  629, 19943, 125456,  224843,  125456,  19943,   629,    1;
  1, 1268, 65171, 647299, 1899096, 1899096, 647299, 65171, 1268, 1;

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Cf. A001053 (row sums), A094002 (column k=2).

Cf. A144431, A144432, A144435, A144436.

T[n_, k_, m_, j_]:= T[n,k,m,j]= If[k==1 || k==n, 1, (m*(n-k)+1)*T[n-1,k-1,m,j] + (m*(k-1)+1)*T[n-1,k,m,j] + j*T[n-2,k-1,m,j]];
Table[T[n,k,1,1], {n,15}, {k,n}]//Flatten (* modified by G. C. Greubel, Mar 03 2022 *)

def T(n,k,m,j):
    if (k==1 or k==n): return 1
    else: return (m*(n-k)+1)*T(n-1,k-1,m,j) + (m*(k-1)+1)*T(n-1,k,m,j) + j*T(n-2,k-1,m,j)
def A144438(n,k): return T(n,k,1,1)
flatten([[A144438(n,k) for k in (1..n)] for n in (1..15)]) # G. C. Greubel, Mar 03 2022

T(n,k) = (n-k+1)*T(n-1, k-1) + k*T(n-1, k) + T(n-2, k-1), T(n, 1) = T(n, n) = 1.
Sum_{k=1..n} T(n, k) = A001053(n+1).
From G. C. Greubel, Mar 03 2022: (Start)
T(n, n-k) = T(n, k).
T(n, 3) = (1/2)*(n^2 +3*n +1 + 73*3^(n-3) - 5*2^(n-2)*(2*n+3)). (End)

A188589 Expansion of (1-3x+6x^2-3x^3)/((1-x)^2(1-2*x)).

Original entry on oeis.org

1, 1, 5, 14, 33, 72, 151, 310, 629, 1268, 2547, 5106, 10225, 20464, 40943, 81902, 163821, 327660, 655339, 1310698, 2621417, 5242856, 10485735, 20971494, 41943013, 83886052, 167772131, 335544290, 671088609, 1342177248, 2684354527

Offset: 0

Paul Barry, Apr 04 2011

Second column of the 1-Euler triangle A188587. In general, the second column of the r-Euler triangle has g.f. (1-(4-r)*x+2*(4-r)*x^2-(4-r)*x^3)/((1-x)^2*(1-2*x)).

Index entries for linear recurrences with constant coefficients, signature (4,-5,2).

CoefficientList[Series[(1-3x+6x^2-3x^3)/((1-x)^2(1-2x)),{x,0,30}],x] (* or *) LinearRecurrence[{4,-5,2},{1,1,5,14},40] (* Harvey P. Dale, Nov 26 2017 *)

a(n) = if (n==0, 1, 5*2^(n-1) - n - 3) \\ Michel Marcus, Jul 24 2013

a(n+1)=A094002(n).

a(n) = 5*2^(n-1)-n-3, n>0.

A336014 Irregular triangle read by rows: T(n,1) = T(n,2) = T(n,3n-2) = T(n,3n-1) = n for n >= 1 and T(n,k) = T(n-1,k-2) + T(n-1,k-1) for n > 1, 3 <= k <= 3*(n-1).

Original entry on oeis.org

1, 1, 2, 2, 2, 2, 2, 3, 3, 4, 4, 4, 4, 3, 3, 4, 4, 6, 7, 8, 8, 8, 7, 6, 4, 4, 5, 5, 8, 10, 13, 15, 16, 16, 15, 13, 10, 8, 5, 5, 6, 6, 10, 13, 18, 23, 28, 31, 32, 31, 28, 23, 18, 13, 10, 6, 6, 7, 7, 12, 16, 23, 31, 41, 51, 59, 63, 63, 59, 51, 41, 31, 23, 16, 12, 7, 7

Offset: 1

Lechoslaw Ratajczak, Jul 04 2020

The number of terms in row n is 3*n-1 = A016789(n-1).
The sum of row n is equal to 2*A094002(n-1) = 2*A188589(n).
Fibonacci(n) = T(n+k,n) - T(n+k-1,n) for n >= 1, k = 1,2,3,...
The elements b(k) of the main diagonal, superdiagonal 1 and all subdiagonals have the recursive formula: b(k) = 2*b(k-1) + b(k-2) - 2*b(k-3) - b(k-4) for k > 4.

			Triangle begins:
n\k 1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20...
1   1  1
2   2  2  2  2  2
3   3  3  4  4  4  4  3  3
4   4  4  6  7  8  8  8  7  6  4  4
5   5  5  8 10 13 15 16 16 15 13 10  8  5  5
6   6  6 10 13 18 23 28 31 32 31 28 23 18 13 10  6  6
7   7  7 12 16 23 31 41 51 59 63 63 59 51 41 31 23 16 12  7  7
...

Superdiagonal 1 is A029907 for n >= 1.
The main diagonal is A208354 for n >= 1.
Subdiagonal 1 is A102702(n-1) for n >= 1.
Subdiagonal 2 is A206268(n+2) for n >= 1 (conjectured).
Subdiagonal 3 is A191830(n+3) for n >= 1.
Cf. A007318, A051597, A228196.

T(n,k) = T(n,3*k-n) for 1 <= k <= 3*n-1.
T(n,k) = Sum_{u=2*(n-k)+3..2*n-k+1} ceiling(u/2)*A065941(k-2,u-2*(n-k)-3) for n >= 3, 3 <= k <= n.
T(n,k) = Sum_{m1=1..k-n} A208354(m1)*binomial(n-m1-1, k-n-m1) + Sum_{m2=1..2*n-k} A208354(m2)*binomial(n-m2-1, 2*n-k-m2) for n >= 2, n+1 <= k <= 2*n-1.
T(n,k) = Sum_{u=2*(k-2*n)+3..k-n+1} ceiling(u/2)*A065941(3*n-k-2,u-2*(k-2*n)-3) for n>= 3, 2*n <= k <= 3*(n-1).
T(n,k) = A208354(k) + (n-k)*Fibonacci(k) for n >= 3, 3 <= k <= n.
T(n,k) = A029907(k-1) + (n-k+1)*Fibonacci(k) for n >= 2, 3 <= k <= n+1.

cp's OEIS Frontend

A157207 Triangle T(n, k, m) = (m*(n-k) + 1)*T(n-1, k-1, m) + (m*k + 1)*T(n-1, k, m) + m*f(n,k)*T(n-2, k-1, m) with T(n, 0, m) = T(n, n, m) = 1, f(n, k) = k if k <= floor(n/2) otherwise n-k, and m = 1, read by rows.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Sage

Formula

Extensions

A274835 Number A(n,k) of set partitions of [n] such that the difference between each element and its block index is a multiple of k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A144438 Triangle T(n,k) by rows: T(n, k) = (n-k+1)*T(n-1, k-1) + k*T(n-1, k) + T(n-2, k-1) with T(n, 1) = T(n, n) = 1.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Sage

Formula

A188589 Expansion of (1-3*x+6*x^2-3*x^3)/((1-x)^2*(1-2*x)).

Views

Author

Keywords

Comments

Links

Programs

Mathematica

PARI

Formula

A336014 Irregular triangle read by rows: T(n,1) = T(n,2) = T(n,3*n-2) = T(n,3*n-1) = n for n >= 1 and T(n,k) = T(n-1,k-2) + T(n-1,k-1) for n > 1, 3 <= k <= 3*(n-1).

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A157207 Triangle T(n, k, m) = (m(n-k) + 1)T(n-1, k-1, m) + (mk + 1)T(n-1, k, m) + mf(n,k)T(n-2, k-1, m) with T(n, 0, m) = T(n, n, m) = 1, f(n, k) = k if k <= floor(n/2) otherwise n-k, and m = 1, read by rows.

A144438 Triangle T(n,k) by rows: T(n, k) = (n-k+1)T(n-1, k-1) + kT(n-1, k) + T(n-2, k-1) with T(n, 1) = T(n, n) = 1.

A188589 Expansion of (1-3x+6x^2-3x^3)/((1-x)^2(1-2*x)).

A336014 Irregular triangle read by rows: T(n,1) = T(n,2) = T(n,3n-2) = T(n,3n-1) = n for n >= 1 and T(n,k) = T(n-1,k-2) + T(n-1,k-1) for n > 1, 3 <= k <= 3*(n-1).