A023531 -id:A023531 - cp's OEIS frontend

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).

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

A370347 Number T(n,k) of partitions of [3n] into n sets of size 3 having exactly k sets {3j-2,3j-1,3j} (1<=j<=n); triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 9, 0, 1, 252, 27, 0, 1, 14337, 1008, 54, 0, 1, 1327104, 71685, 2520, 90, 0, 1, 182407545, 7962624, 215055, 5040, 135, 0, 1, 34906943196, 1276852815, 27869184, 501795, 8820, 189, 0, 1, 8877242235393, 279255545568, 5107411260, 74317824, 1003590, 14112, 252, 0, 1

Offset: 0

Alois P. Heinz, Feb 15 2024

			T(2,0) = 9: 124|356, 125|346, 126|345, 134|256, 135|246, 136|245, 145|236, 146|235, 156|234.
T(2,2) = 1: 123|456.
Triangle T(n,k) begins:
            1;
            0,          1;
            9,          0,        1;
          252,         27,        0,      1;
        14337,       1008,       54,      0,    1;
      1327104,      71685,     2520,     90,    0,   1;
    182407545,    7962624,   215055,   5040,  135,   0, 1;
  34906943196, 1276852815, 27869184, 501795, 8820, 189, 0, 1;
  ...

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

Row sums give A025035.
Column k=0 gives A370357.
T(n+1,n-1) gives A027468.
T(n+2,n-1) gives 252*A000292.
Cf. A023531, A055140, A370358.

b:= proc(n) option remember; `if`(n<3, [1, 0, 9][n+1],
      9*(n*(n-1)/2*b(n-1)+(n-1)^2*b(n-2)+(n-1)*(n-2)/2*b(n-3)))
    end:
T:= (n, k)-> b(n-k)*binomial(n, k):
seq(seq(T(n, k), k=0..n), n=0..10);

T(n,k) = binomial(n,k) * A370357(n-k).
Sum_{k=1..n} T(n,k) = A370358(n).
T(n,k) mod 9 = A023531(n,k).

A168313 Triangle read by rows, retain 1's as rightmost diagonal of A101688 and replace all other 1's with 2's.

Original entry on oeis.org

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

Offset: 1

Gary W. Adamson, Nov 22 2009

Row sums = odd integers repeated: (1, 1, 3, 3, 5, 5,...).

Eigensequence of the triangle = A168314: (1, 1, 3, 5, 13, 29, 71, 165, 401,...).

			First few rows of the triangle =
1;
0, 1;
0, 2, 1;
0, 0, 2, 1;
0, 0, 2, 2, 1;
0, 0, 0, 2, 2, 1;
0, 0, 0, 2, 2, 2, 1;
0, 0, 0, 0, 2, 2, 2, 1;
0, 0, 0, 0, 2, 2, 2, 2, 1;
0, 0, 0, 0, 0, 2, 2, 2, 2, 1;
0, 0, 0, 0, 0, 2, 2, 2, 2, 2, 1;
0, 0, 0, 0, 0, 0, 2, 2, 2, 2, 2, 1;
...

Boris Putievskiy, Transformations (of) Integer Sequences And Pairing Functions, arXiv:1212.2732 [math.CO], 2012.

Cf. A101688, A168314, A168315

rows = 11;
A = Array[Which[#1 == 1, 1, #1 <= #2, 2, True, 0]&, {rows, rows}];
Table[A[[i-j+1, j]], {i, 1, rows}, {j, 1, i}] // Flatten (* Jean-François Alcover, Aug 08 2018 *)

Triangle read by rows, retain 1's as rightmost diagonal of A101688 and replace all other 1's with 2's.
From Boris Putievskiy, Jan 09 2013: (Start)
a(n) = 2*A101688(n)-A023531(n).
a(n) = 2*floor((2*A002260(n)+1)/(A003056(n)+3))*A002260(n)-A023531(n).
a(n) = 2*floor((2*n-t*(t+1)+1)/(t+3))*(n-t*(t+1)/2) - floor((sqrt(8*n+1)-1)/2) + t, where t = floor((-1+sqrt(8*n-7))/2). (End)

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).

A318147 Coefficients of the Omega polynomials of order 3, triangle T(n,k) read by rows with 0<=k<=n.

Original entry on oeis.org

1, 0, 1, 0, -9, 10, 0, 477, -756, 280, 0, -74601, 142362, -83160, 15400, 0, 25740261, -55429920, 40900860, -12612600, 1401400, 0, -16591655817, 38999319642, -33465991104, 13440707280, -2572970400, 190590400

Offset: 0

Peter Luschny, Aug 22 2018

The name 'Omega polynomial' is not a standard name.

			[0] [1]
[1] [0,            1]
[2] [0,           -9,          10]
[3] [0,          477,        -756,          280]
[4] [0,       -74601,      142362,       -83160,       15400]
[5] [0,     25740261,   -55429920,     40900860,   -12612600,     1401400]
[6] [0, -16591655817, 38999319642, -33465991104, 13440707280, -2572970400,190590400]

All row sums are 1, alternating row sums (taken absolute) are A002115.
T(n,1) ~ A293951(n), T(n,n) = A025035(n).
A023531 (m=1), A318146 (m=2), this seq (m=3), A318148 (m=4).

# See A318146 for the missing functions.
FL([seq(CL(OmegaPolynomial(3, n)), n=0..8)]);

(* OmegaPolynomials are defined in A318146 *)
Table[CoefficientList[OmegaPolynomial[3, n], x], {n, 0, 6}] // Flatten

# See A318146 for the function OmegaPolynomial.
[list(OmegaPolynomial(3, n)) for n in (0..6)]

Omega(m, n, z) = (m*n)!*[z^(n*m)] H(m, z)^x where H(m, z) = hypergeom([], [seq(i/m, i=1..m-1)], (z/m)^m). We consider here the case m = 3 (for other cases see the cross-references).

A318148 Coefficients of the Omega polynomials of order 4, triangle T(n,k) read by rows with 0<=k<=n.

Original entry on oeis.org

1, 0, 1, 0, -34, 35, 0, 11056, -16830, 5775, 0, -14873104, 27560780, -15315300, 2627625, 0, 56814228736, -119412815760, 84786627900, -24734209500, 2546168625, 0, -495812444583424, 1140896479608800, -948030209181000, 364143337057500, -65706427536750, 4509264634875

Offset: 0

Peter Luschny, Aug 22 2018

The name 'Omega polynomial' is not a standard name.

			[0] [1]
[1] [0,           1]
[2] [0,         -34,            35]
[3] [0,       11056,        -16830,        5775]
[4] [0,   -14873104,      27560780,   -15315300,      2627625]
[5] [0, 56814228736, -119412815760, 84786627900, -24734209500, 2546168625]

All row sums are 1, alternating row sums (taken absolute) are A211212.
T(n,1) ~ A273352(n), T(n,n) = A025036(n).
A023531 (m=1), A318146 (m=2), A318147 (m=3), this seq (m=4).

# See A318146 for the missing functions.
FL([seq(CL(OmegaPolynomial(4, n)), n=0..8)]);

(* OmegaPolynomials are defined in A318146 *)
Table[CoefficientList[OmegaPolynomial[4, n], x], {n, 0, 6}] // Flatten

# See A318146 for the function OmegaPolynomial.
[list(OmegaPolynomial(4, n)) for n in (0..6)]

Omega(m, n, z) = (m*n)!*[z^(n*m)] H(m, z)^x where H(m, z) = hypergeom([], [seq(i/m, i=1..m-1)], (z/m)^m). We consider here the case m = 4 (for other cases see the cross-references).

A363732 Triangle read by rows. The triangle algorithm applied to (-1)^n/n!.

Original entry on oeis.org

1, -2, 1, 5, -4, 1, -15, 15, -6, 1, 52, -60, 30, -8, 1, -203, 260, -150, 50, -10, 1, 877, -1218, 780, -300, 75, -12, 1, -4140, 6139, -4263, 1820, -525, 105, -14, 1, 21147, -33120, 24556, -11368, 3640, -840, 140, -16, 1, -115975, 190323, -149040, 73668, -25578, 6552, -1260, 180, -18, 1

Offset: 0

Peter Luschny, Jun 18 2023

The triangle algorithm, as understood here, is a transformation that maps a sequence of integers (a(n) : n >= 0) to a polynomial sequence. A polynomial sequence is a sequence of polynomials (P(n,x) : n >= 0) with degree(P(n, x)) = n for all n >= 0.
The polynomials P(n, x) are recursively defined by P(n, x) = p(n, 0, x), where the initial sequence is p(0, m, x) = a(m), and for n > 0 is given by
  p(n, m, x) = (m + 1)*p(n - 1, m + 1, x) - (m + 1 - x)*p(n - 1, m, x).
Here we identify the polynomial sequence with the infinite lower triangular array of its coefficients, T(n, k) = [x^k] P(n, x). We call the mapping (a(n) : n >= 0) -> (T(n, k) : 0 <= k <= n) the 'triangle algorithm', following the lead of Kawasaki and Ohno.
Evaluating P(n, x) at different values of x gives rise to a multitude of other sequences; in particular, the transformation a(n) -> b(n) = P(n, 1) will be called the Akiyama-Tanigawa transform of a.
The triangle algorithm was studied by Akiyama and Tanigawa, Chen, Imatomi, Arakawa and Kaneko, Kawasaki and Ohno, and others, at first in connection with the Bernoulli and Poly-Bernoulli numbers.
.
The paradigmatic examples are:
  a(n) = 1 -> x^n, the standard base of polynomials, A023531.
  a(n) = n + 1 -> binomial(n, k), Pascal triangle, A007318.
  a(n) = n + 1 -> P(n, 1) powers of 2, A000079.
  a(n) = n + 1 -> P(n, 0) the all 1's sequence A000012.
  a(n) = 2^n -> [x^k] P(n, x), A154921.
  a(n) = 2^n -> P(n, 0) Fubini numbers, A000670.
  a(n) = 2^n -> P(n, 1) ordered set partitions of subsets of [n], A000629.
  a(n) = 2^n -> P(n,-1) osp. of [n] with even number of blocks, A052841.
  a(n) = 1 / (n + 1) -> [x^k] B(n, x), Bernoulli polynomials, A196838/A196839.
  a(n) = 1 / (n + 1) -> B(n, 1), the Bernoulli numbers, A164555/A027642.
  a(n) = Chen(n) -> skp(n, x), Swiss-Knife polynomials, A153641.
  a(n) = Chen(n) -> P(n, 0), 2^n*Euler(n, 1/2) = Euler(n), A122045.
  a(n) = Chen(n) -> P(n, 1), 2^n*Euler(n, 1), A155585.
  a(n) = (-1)^n/n! -> [x^k] P(n, x) this "Bell" triangle.
  a(n) = (-1)^n/n! -> (-1)^n*P(n, 1) = Bell(n), A000110.
  a(n) = (-1)^n/n! -> (-1)^n*P(n,-1) = 2-Bell(n), A005493.
  a(n) = 1/n! -> (-1)^n*P(n, 1) = complementary Bell(n), A000587.
  a(n) = 1/n! -> (-1)^n*P(n,-1) = complementary 2-Bell(n), A074051.
  (For Chen's sequence see A363524.)
.
The present sequence deals with the case of the Bell numbers. In contrast to Aitken's array A011971 and its variants A123346 and A011972, the Bell numbers do not appear as a column of the triangle but as row sums (times (-1)^n), i.e., as values of the associated polynomials at x = 1. Comparing this with a similar situation with the Bernoulli numbers/polynomials, our triangle could be viewed as a more organic generalization of the Bell numbers. Indeed, the names 'Bell triangle' and 'Bell polynomials' would be justified here; but these are already assigned to other concepts.

			The triangle T(n, k) starts:
  [0]     1;
  [1]    -2,      1;
  [2]     5,     -4,     1;
  [3]   -15,     15,    -6,      1;
  [4]    52,    -60,    30,     -8,    1;
  [5]  -203,    260,  -150,     50,  -10,    1;
  [6]   877,  -1218,   780,   -300,   75,  -12,   1;
  [7] -4140,   6139, -4263,   1820, -525,  105, -14,   1;
  [8] 21147, -33120, 24556, -11368, 3640, -840, 140, -16, 1;

Peter Luschny, Table of n, a(n) for row 0..100.
Kwang-Wu Chen, Algorithms for Bernoulli numbers and Euler numbers, J. Integer Sequences, 4 (2001), #01.1.6.
Masanobu Kaneko, The Akiyama-Tanigawa algorithm for Bernoulli numbers, J. Integer Sequences, 3 (2000), #00.2.9.
Naho Kawasaki and Yasuo Ohno, The triangle algorithm for Bernoulli polynomials, Integers, vol. 23 (2023). (See figure 4.)
D. Merlini, R. Sprugnoli, and M. C. Verri, The Akiyama-Tanigawa Transformation, Integers, 5 (1) (2005) #A05.

Cf. A293037 (row sums), A000110 (row sums, unsigned), A005493 (alternating row sums, signed).

Cf. A023531, A007318, A000079, A000012, A154921, A000670, A196838/A196839, A164555/A027642, A153641, A122045, A155585, A011971, A123346, A011972, A056857, A363524 (Chen sequence).

TA := proc(a, n, m, x) option remember; if n = 0 then a(m) else
normal((m + 1)*TA(a, n - 1, m + 1, x) - (m + 1 - x)*TA(a, n - 1, m, x)) fi end:
seq(seq(coeff(TA(n -> (-1)^n/n!, n, 0, x), x, k), k = 0..n), n = 0..10);

(* rows[0..n], n[0..oo] *)
(* row[n]= *)
n=9;r={};For[a=n+1,a>0,a--,AppendTo[r,(-1)^(a+1)*Sum[StirlingS2[a,k],{k,0,a}]*Product[(2*(a+j))/(2*j+2),{j,0,n-a}]]];r
(* columns[1..n], n[0..oo] *)
(* column[n]= *)
n=0;c={};For[a=1,a<15,a++,AppendTo[c,(-1)^(a+1)*Sum[StirlingS2[a,k],{k,0,a}]*Product[(2*(a+j-1))/(2*j),{j,1,n}]]];c
(* sequence *)
s={};For[n=0,n<15,n++,For[a=n+1,a>0,a--,AppendTo[s,(-1)^(a+1)*Sum[StirlingS2[a,k],{k,0,a}]*Product[(2*(a+j))/(2*j+2),{j,0,n-a}]]]];s
(* Detlef Meya, Jun 22 2023 *)

def a(n): return (-1)^n / factorial(n)
@cached_function
def p(n, m):
    R = PolynomialRing(QQ, "x")
    if n == 0: return R(a(m))
    return R((m + 1)*p(n - 1, m + 1) - (m + 1 - x)*p(n - 1, m))
for n in range(10): print(p(n, 0).list())

A023604 Convolution of A023532 and A023533.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A023531, A023532, A023533.

A023532:= func< n | IsIntegral((Sqrt(8*n+9) - 3)/2) select 0 else 1 >;
A023533:= func< n | Binomial(Floor((6*n-1)^(1/3)) +2, 3) ne n select 0 else 1 >;
[(&+[A023533(k)*A023532(n+1-k): k in [1..n]]): n in [1..100]]; // G. C. Greubel, Jul 16 2022

A023533[n_]:= If[Binomial[Floor[Surd[6*n-1, 3]] +2, 3] != n, 0, 1];
A023531[n_]:= If[IntegerQ[(Sqrt[8*n+9] -3)/2], 1, 0];
A023604[n_]:= A023604[n]= Sum[A023533[k]*(1-A023531[n-k+1]), {k,n}];
Table[A023604[n], {n,100}] (* G. C. Greubel, Jul 16 2022 *)

def A023532(n): return 0 if ((sqrt(8*n+9) -3)/2).is_integer() else 1
@CachedFunction
def A023533(n): return 0 if (binomial( floor( (6*n-1)^(1/3) ) +2, 3) != n) else 1
[sum(A023533(k)*A023532(n-k+1) for k in (1..n)) for n in (1..100)] # G. C. Greubel, Jul 16 2022

From G. C. Greubel, Jul 16 2022: (Start)
a(n) = Sum_{j=1..n} A023532(n-j+1) * A023533(j).
a(n) = Sum_{j=1..n} (1 - A023531(n-j+1)) * A023533(j). (End)

A049323 Triangle of coefficients of certain polynomials (exponents in increasing order), equivalent to A033842.

Original entry on oeis.org

1, 1, 1, 1, 3, 3, 1, 6, 16, 16, 1, 10, 50, 125, 125, 1, 15, 120, 540, 1296, 1296, 1, 21, 245, 1715, 7203, 16807, 16807, 1, 28, 448, 4480, 28672, 114688, 262144, 262144, 1, 36, 756, 10206, 91854, 551124, 2125764, 4782969, 4782969, 1, 45, 1200, 21000, 252000

Offset: 0

Wolfdieter Lang

These polynomials p(n, x) appear in the W. Lang reference as c1(-(n+1);x), n >= 0 on p.12. The coefficients are given there in eq.(44) on p. 6. - Wolfdieter Lang, Nov 20 2015

			The triangle a(n, m) begins:
n\m 0  1   2    3     4      5      6      7 ...
0:  1
1:  1  1
2:  1  3   3
3:  1  6  16   16
4:  1 10  50  125  125
5:  1 15 120  540  1296  1296
6:  1 21 245 1715  7203  16807  16807
7:  1 28 448 4480 28672 114688 262144 262144
... reformatted. - Wolfdieter Lang, Nov 20 2015
E.g. the third row {1,3,3} corresponds to polynomial p(2,x)= 1 + 3*x + 3*x^2.

W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

a(n, 0)= A000012 (powers of 1), a(n, 1)= A000217 (triangular numbers), a(n, n)= A000272(n+1), n >= 0 (diagonal), a(n, n-1)= A000272(n+1), n >= 1.
For n = 0..5 the row sequences a(n, m), m >= 0, are the first columns of the triangles A023531 (unit matrix), A030528, A049324, A049325, A049326, A049327, respectively.
Cf. A033842, A046757.

/* As triangle: */ [[Binomial(n+1, k+1)*(n+1)^(k-1): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Nov 20 2015

seq(seq(binomial(n+1,m+1)*(n+1)^(m-1),m=0..n),n=0..10); # Robert Israel, Oct 19 2015

Table[Binomial[n + 1, k + 1] (n + 1)^(k - 1), {n, 0, 9}, {k, 0, n}] // Flatten (* Michael De Vlieger, Nov 19 2015 *)

a(n, m) = A033842(n, n-m) = binomial(n+1, m+1)*(n+1)^{m-1}, n >= m >= 0, else 0.
p(k-1, -x)/(1-k*x)^k =(-1+1/(1-k*x)^k)/(x*k^2) is for k=1..5 G.f. for A000012, A001792, A036068, A036070, A036083, respectively.
From Werner Schulte, Oct 19 2015: (Start)
a(2*n,n) = A000108(n)*(2*n+1)^n;
a(3*n,2*n) = A001764(n)*(3*n+1)^(2*n);
a(p*n,(p-1)*n) = binomial(p*n,n)/((p-1)*n+1)*(p*n+1)^((p-1)*n) for p > 0;
Sum_{m=0..n} (m+1)*a(n,m) = (n+2)^n;
Sum_{m=0..n} (-1)^m*(m+1)*a(n,m) = (-n)^n where 0^0 = 1;
p(n,x) = Sum_{m=0..n} a(n,m)*x^m = ((1+(n+1)*x)^(n+1)-1)/((n+1)^2*x).
(End)

cp's OEIS Frontend

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