A011971 -id:A011971 - cp's OEIS frontend

A259697 Triangle read by rows: T(n,k) = number of rook patterns on n X n board where bottom rook is in column k.

1, 1, 1, 1, 2, 2, 1, 4, 5, 5, 1, 9, 12, 15, 15, 1, 24, 32, 42, 52, 52, 1, 76, 99, 129, 166, 203, 203, 1, 279, 354, 451, 575, 726, 877, 877, 1, 1156, 1434, 1786, 2232, 2792, 3466, 4140, 4140, 1, 5296, 6451, 7883, 9664, 11881, 14621, 17884, 21147, 21147

Offset: 0

N. J. A. Sloane, Jul 05 2015

See Becker (1948/49) for precise definition.

This is the number of arrangements of non-attacking rooks on an n X n right triangular board where the bottom rook is in column k. The case of k=0 corresponds to the empty board where there is no bottom rook. - Andrew Howroyd, Jun 13 2017

			Triangle begins:
  1,
  1,  1,
  1,  2,  2,
  1,  4,  5,   5,
  1,  9, 12,  15,  15,
  1, 24, 32,  42,  52,  52,
  1, 76, 99, 129, 166, 203, 203,
  ...
From _Andrew Howroyd_, Jun 13 2017: (Start)
For n=3, the four solutions with the bottom rook in column 1 are:
  .      .      .      x
  . .    . x    x .    . .
  x . .  x . .  . . .  . . .
For n=3, the five solutions with the bottom rook in column 2 are:
  .      .      x      .       x
  . .    x .    . .    . x     . x
  . x .  . x .  . x .  . . .   . . .
(End)

Andrew Howroyd, Table of n, a(n) for n = 0..1274
H. W. Becker, Rooks and rhymes, Math. Mag. 22 (1948/49), 23-26. See Table V.
H. W. Becker, Rooks and rhymes, Math. Mag., 22 (1948/49), 23-26. [Annotated scanned copy]

Right edge is A000110.
Column k=1 is A005001.
Row sums are A000110(n+1).
Cf. A011971, A259691.

a11971[n_, k_] := Sum[Binomial[k, i]*BellB[n - k + i], {i, 0, k}];
T[, 0] = 1; T[n, k_] := Sum[a11971[i - 1, k - 1], {i, k, n}];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 07 2018, after Andrew Howroyd *)

A(N)={my(T=matrix(N,N),U=matrix(N,N));T[1,1]=1;U[1,1]=1;
for(n=2,N,for(k=1,n, T[n,k]=if(k==1,T[n-1,n-1],T[n-1,k-1]+T[n,k-1]); U[n,k]=T[n,k]+U[n-1,k]));U}
{my(T=A(10));for(n=0,length(T),for(k=0,n,print1(if(k==0,1,T[n,k]),", "));print)} \\ Andrew Howroyd, Jun 13 2017

from sympy import bell, binomial
def a011971(n, k): return sum([binomial(k, i)*bell(n - k + i) for i in range(k + 1)])
def T(n, k): return 1 if k==0 else sum([a011971(i - 1, k - 1) for i in range(k, n + 1)])
for n in range(10): print([T(n, k) for k in range(n + 1)]) # Indranil Ghosh, Jun 17 2017

T(n,0) = 1, T(n,k) = Sum_{i=k..n} A011971(i-1,k-1) for k>0. - Andrew Howroyd, Jun 13 2017

Terms a(28) and beyond from Andrew Howroyd, Jun 13 2017

A127741 a(n) = (n+1) * A005493(n).

Original entry on oeis.org

1, 6, 30, 148, 755, 4044, 22841, 136056, 853452, 5625950, 38885297, 281170080, 2122313505, 16688829122, 136457754030, 1158155642512, 10186602918035, 92711977180164, 871936904575985, 8462913158427580, 84668764368102012, 872196382566014506, 9241557859113581689

Offset: 0

Gary W. Adamson, Jan 27 2007

nonn

			a(n) = sum of terms in n-th row of A127740. a(2) = 30 = (6 + 9 + 15).

Chai Wah Wu, Table of n, a(n) for n = 0..250

Cf. A005493, A052889, A011971, A127740 (row sums).

lim=24;A005493=Differences[BellB[Range[lim]]];Array[(#+1)*A005493[[#+1]]&,lim-1,0] (* James C. McMahon, Jan 02 2025 *)

# requires Python 3.2 or higher. Otherwise use def'n of accumulate in Python docs.
from itertools import accumulate
A127741_list, blist, b = [], [1], 1
for n in range(1,1001):
    blist = list(accumulate([b]+blist))
    b = blist[-1]
    A127741_list.append(blist[-2]*n) # Chai Wah Wu, Sep 20 2014

a(n) = (n+1) * A005493(n).

Edited by Jon E. Schoenfield, May 27 2019

A159573 Triangle read by rows, A055248 * (A005493 * 0^(n-k)).

Original entry on oeis.org

1, 2, 1, 4, 3, 3, 8, 7, 12, 10, 16, 15, 33, 50, 37, 32, 31, 78, 160, 222, 151, 64, 63, 171, 420, 814, 1057, 674, 128, 127, 360, 990, 2368, 4379, 5392, 3263, 256, 255, 741, 2190, 6031, 14043, 24938, 29367, 17007, 512, 511, 1506, 4660, 14134, 38656, 87620

Offset: 0

Gary W. Adamson, Apr 16 2009

Row sums = A005493: (1, 3, 10, 37, 151, 674, 3263,...); = row sums of Aitken's array. As a property of eigentriangles, sum of n-th row terms = rightmost term of next row.

			First few rows of the triangle =
1;
2, 1;
4, 3, 3;
8, 7, 12, 10;
16, 15, 33, 50, 37;
32, 31, 78, 160, 222, 151;
64, 63, 171, 420, 814, 1057, 674;
128, 127, 360, 990, 2368, 4379, 5392, 3263;
256, 255, 741, 2190, 6031, 14043, 24938, 29367, 17007;
512, 511, 1506, 4660, 14134, 38656, 87620, 150098, 170070, 94828;
1024, 1023, 3039, 9680, 31376, 96338, 260164, 574288, 952392, 1043108, 562595;
...
Example: row 3 = (8, 7, 12, 10) = termwise products of (8, 7, 4, 1) and
(1, 1, 3, 10), where (8, 7, 12, 10) = row 3 of triangle A055248.

Cf. A055248, A005493, A011971

Triangle read by rows, A055248 * (A005493 * 0^(n-k)); equivalent to the product of triangle A055248 and its own eigensequence (diagonalized with the rest zeros, as an infinite lower triangular matrix).

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())

A105605 An Aitken-like array: see Comments for definition.

Original entry on oeis.org

1, 1, 2, 2, 3, 5, 5, 7, 1, 0, 6, 6, 1, 1, 8, 2, 8, 8, 8, 1, 4, 2, 5, 1, 0, 7, 9, 8, 8, 1, 6, 2, 1, 0, 4, 6, 1, 4, 1, 3, 1, 0, 1, 2, 2, 1, 0, 2, 6, 4, 7, 4, 1, 1, 1, 0, 2, 5, 2, 3, 3, 5, 3, 5, 5, 7, 8, 8, 1, 0, 7, 4, 1, 4, 8, 2, 5, 9, 2, 7, 1, 4, 4, 1, 0, 4, 9, 7, 6, 6, 1, 1, 8, 9, 1, 6, 1, 0, 1

Offset: 1

Eric Angelini, Sep 29 2009

The entries are single digits.
- start the first row with 1
- a new row starts with the last digit of the previous one
- the successive digits on the same row are given by the 'vertical sum' operation applied to the current digit
- 'vertical sum': add to digit 'd' the digit exactly above 'd' and write the result at the right of 'd'
- if the vertical sum exceeds 9, separate the digits with a space
- proceed till the end of the row.

			We see here that the 4th row shows the 'vertical sum' 7+3 = 10 to the right of 7 like this: 7,1 0.
The array begins:
1.
1,2.
2,3,5.
5,7,1 0,6.
6,1 1,8,2,8,8.
8,1 4,2,5,1 0,7,9,8.
8,1 6,2,1 0,4,6,1,4,1 3,1 0,1 2.
2,1 0,2,6,4,7,4,1 1,1 0,2,5,2,3,3,5,3,5.
5,7,8,8,1 0,7,4,1 4,8,2,5,9,2,7,1 4,4,1 0,4,9,7,6.
6,1 1,8,9,1 6,1 0,1,1 3,5,1,5,9,5,1 0,1 0,7,1 6,6,5,4,2,0,1 1,1 0,1 3,1 2.
2,8,9,1 0,9,9,1 0,1 5,2,0,2,6,5,5,3,1 1,1 4,1 0,4,1,2,1,1 1,2,6,1 0,6,6,3,1,2,3,7,1,1,9,7,5.
5,7,1 5,1 0,6,1,9,1 5,2,9,2,1 0,4,9,4,7,5,9,1 2,5,8,6,1 3,2,2,9,9,8,2,4,3,4,1 5,1 0,8,8,1 0,6,5,3,8,8,1,9,1 7,8,5.
5,1 0,8,1,1 3,2,1,9,3,1 0,1 0,8,3,9,3,1,8,7,1 8,7,8,1 3,1 6,2,1 0,1 2,1 6,7,4,4,8,4,1 0,9,9,4,5,9,1 1,5,9,9,4,9,8,1 0,...

Cf. A011971, A126971.

A127740 Natural number transform of Aitken's triangle.

Original entry on oeis.org

1, 2, 4, 6, 9, 15, 20, 28, 40, 60, 75, 100, 135, 185, 260, 312, 402, 522, 684, 906, 1218, 1421, 1785, 2254, 2863, 3661, 4718, 6139, 7016, 8640, 10680, 13256, 16528, 20712, 26104, 33120, 37260, 45153, 54873, 66888, 81801, 100395, 123696, 153063, 190323

Offset: 0

Gary W. Adamson, Jan 27 2007

Left column (1, 2, 6, 20, ...) = A052889.

Row sums give A127741.

			First few rows of the triangle:
   1;
   2,   4;
   6,   9,  15;
  20,  28,  40,  60;
  75, 100, 135, 185, 260;
  ...

Cf. A005493, A052889, A011971, A127741.

A127648 * A011971 as infinite lower triangular matrices.

A247108 Complementary Aitken's array: triangle of numbers {a(n,k), n >= 0, 0<=k<=n} read by rows, defined by a(0,0)=1, a(n,0)=-a(n-1,n-1), a(n,k)=a(n,k-1)+a(n-1,k-1).

Original entry on oeis.org

1, -1, 0, 0, -1, -1, 1, 1, 0, -1, 1, 2, 3, 3, 2, -2, -1, 1, 4, 7, 9, -9, -11, -12, -11, -7, 0, 9, -9, -18, -29, -41, -52, -59, -59, -50, 50, 41, 23, -6, -47, -99, -158, -217, -267, 267, 317, 358, 381, 375, 328, 229, 71, -146, -413, 413, 680, 997, 1355, 1736, 2111, 2439, 2668, 2739, 2593, 2180

Offset: 0

Chai Wah Wu, Nov 19 2014

a(n,0) of the triangle is equal to A000587(n), the Rao Uppuluri-Carpenter numbers or complementary Bell numbers.

			Triangle begins:
00:      1
01:     -1      0
02:      0     -1     -1
03:      1      1      0     -1
04:      1      2      3      3      2
05:     -2     -1      1      4      7      9
06:     -9    -11    -12    -11     -7      0      9
07:     -9    -18    -29    -41    -52    -59    -59    -50
08:     50     41     23     -6    -47    -99   -158   -217   -267
09:    267    317    358    381    375    328    229     71   -146   -413

Chai Wah Wu, Rows n=0..50 of triangle, flattened
D. Wuilquin, Letters to N. J. A. Sloane, August 1984

Cf. A000587, A011971, A000110.

a247108 n k = a247108_tabl !! n !! k
a247108_row n = a247108_tabl !! n
a247108_tabl = iterate (\row -> scanl (+) (- last row) row) [1]
-- Reinhard Zumkeller, Nov 22 2014

a[0, 0] = 1;
a[n_, 0] := -a[n - 1, n - 1];
a[n_, k_] /; 0 <= k <= n := a[n, k] = a[n, k - 1] + a[n - 1, k - 1];
a[, ] = 0;
Table[a[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 20 2019 *)

from itertools import accumulate
A247108_list = blist = [1]
for _ in range(10**2):
    b = -blist[-1]
    blist = list(accumulate([b]+blist))
    A247108_list += blist

A280470 Triangle A106534 with reversed rows.

Original entry on oeis.org

1, 1, 2, 2, 3, 5, 5, 7, 10, 15, 14, 19, 26, 36, 51, 42, 56, 75, 101, 137, 188, 132, 174, 230, 305, 406, 543, 731, 429, 561, 735, 965, 1270, 1676, 2219, 2950, 1430, 1859, 2420, 3155, 4120, 5390, 7066, 9285, 12235, 4862, 6292, 8151, 10571, 13726, 17846, 23236, 30302, 39587, 51822, 16796, 21658, 27950, 36101, 46672

Offset: 0

Tony Foster III, Jan 03 2017

			Fibonacci Determinant Triangle:
    1;
    1,    2;
    2,    3,    5;
    5,    7,   10,   15;
   14,   19,   26,   36,   51;
   42,   56,   75,  101,  137,  188;
  132,  174,  230,  305,  406,  543,  731;
  429,  561,  735,  965, 1270, 1676, 2219, 2950;
  ...

P. Barry, A. Hennessy, The Euler-Seidel Matrix, Hankel Matrices and Moment Sequences, J. Int. Seq. 13 (2010) # 10.8.2, page 5
A. Cvetkovi, Predrag Rajkovic, and Milos IvkoviCatalan Numbers, the Hankel Transform, and Fibonacci Numbers, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.3.

Cf. A000108, A001519, A007317, A011971, A103433, A106534, A197649.

&cat [[&+[Binomial(k,j)*Catalan(n-j): j in [0..k]]: k in [0..n]]: n in [0..10]]; // Bruno Berselli, Mar 07 2017

Table[Sum[Binomial[k, j] CatalanNumber[n - j], {j, 0, k}], {n, 0, 10}, {k, 0, n}] // Flatten (* Michael De Vlieger, Mar 08 2017 *)

C(n)=binomial(2*n,n)/(n+1);
T(n,k)=sum(j=0,k,binomial(k,j)*C(n-j));
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print()); \\ Joerg Arndt, Jan 15 2017

T(n,k) = Sum_{j=0..k} binomial(k,j) * A000108(n-j). - Joerg Arndt, Jan 15 2017

A376177 Triangle, read by rows, where T(n,k) = T(n-1,k-1) + 2*T(n,k-1) when k > 0, else T(n,0) = T(n-1,n-1) when n > 0, with T(0,0) = 1.

Original entry on oeis.org

1, 1, 3, 3, 7, 17, 17, 37, 81, 179, 179, 375, 787, 1655, 3489, 3489, 7157, 14689, 30165, 61985, 127459, 127459, 258407, 523971, 1062631, 2155427, 4372839, 8873137, 8873137, 17873733, 36005873, 72535717, 146134065, 294423557, 593219953, 1195313043, 1195313043, 2399499223, 4816872179, 9669750231, 19412036179, 38970206423, 78234836403, 157062892759, 315321098561, 315321098561

Offset: 0

George Plousos and Paul D. Hanna, Sep 22 2024

This triangle was found by George Plousos while exploring a variation of Aitken's array (A011971).

			G.f.: A(x,y) = 1 + (3*y + 1)*x + (17*y^2 + 7*y + 3)*x^2 + (179*y^3 + 81*y^2 + 37*y + 17)*x^3 + (3489*y^4 + 1655*y^3 + 787*y^2 + 375*y + 179)*x^4 + (127459*y^5 + 61985*y^4 + 30165*y^3 + 14689*y^2 + 7157*y + 3489)*x^5 + (8873137*y^6 + 4372839*y^5 + 2155427*y^4 + 1062631*y^3 + 523971*y^2 + 258407*y + 127459)*x^6 + ...
which is defined by A(x,y) = (B(x) - 2*(B(x*y) - 1)/x) / (1 - (2+x)*y),
where B(x) = 1 + x*B( 2*x/(1-x) )/(1-x) is the g.f. B(x) for A126443,
B(x) = 1 + x + 3*x^2 + 17*x^3 + 179*x^4 + 3489*x^5 + 127459*x^6 + 8873137*x^7 + 1195313043*x^8 + 315321098561*x^9 + ... + A126443(n)*x^n + ...
This triangle begins
  1,
  1, 3,
  3, 7, 17,
  17, 37, 81, 179,
  179, 375, 787, 1655, 3489,
  3489, 7157, 14689, 30165, 61985, 127459,
  127459, 258407, 523971, 1062631, 2155427, 4372839, 8873137,
  8873137, 17873733, 36005873, 72535717, 146134065, 294423557, 593219953, 1195313043,
  ...

Paul D. Hanna, Table of n, a(n) for n = 0..1275

Cf. A011971, A126443.

{A126443(n) = if(n==0, 1, sum(k=0, n-1, binomial(n-1, k) * 2^k * A126443(k)))}
{T(n,k) = sum(j=0,k, binomial(k,j) * 2^j * A126443(n-k+j) )}
for(n=0,10, for(k=0,n, print1(T(n,k),", ")); print(""))

If k > 0, T(n,k) = T(n-1,k-1) + 2*T(n,k-1), else if n > 0, T(n,0) = T(n-1,n-1), with T(0,0) = 1.
T(n,k) = Sum_{j=0..k} binomial(k,j) * 2^j * A126443(n-k+j), where A126443(m) = Sum_{k=0..m-1} binomial(m-1, k) * 2^k * A126443(k) for m > 0 with A126443(0) = 1.
G.f. A(x,y) = (B(x) - 2*(B(x*y) - 1)/x) / (1 - (2+x)*y), where B(x) = 1 + x*B( 2*x/(1-x) )/(1-x) is the g.f. B(x) for A126443 given therein by Ilya Gutkovskiy.

A275871 Row sums and second diagonal of A046934.

Original entry on oeis.org

1, 1, 4, 15, 61, 272, 1317, 6865, 38278, 227093, 1426921, 9457918, 65898275, 481177881, 3672102116, 29218285875, 241873478425, 2079079678176, 18524191138689, 170803860905237, 1627465240969382

Offset: 0

Olivier Gérard, Aug 11 2016

The offset corresponds to the definition of A046934.

Differences of A032346 and of A032347.

Cf. A046934, A046935, A032346, A032347.

Cf. A005493 (first differences of Bell numbers, second diagonal and row sum of A011971).

Clear[d]; d[0] = 1; d[1] = 0; d[n_] := d[n] =
  1 + Sum[Binomial[n - 1, j]*d[j], {j, 2, n - 1}]; Table[
d[n + 2] - d[n + 1], {n, 0, 22}] (* From the code by  J.-F. Alcover and Jon Perry in A032347 *)

cp's OEIS Frontend

A259697 Triangle read by rows: T(n,k) = number of rook patterns on n X n board where bottom rook is in column k.

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A127741 a(n) = (n+1) * A005493(n).

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A159573 Triangle read by rows, A055248 * (A005493 * 0^(n-k)).

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

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A105605 An Aitken-like array: see Comments for definition.

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A127740 Natural number transform of Aitken's triangle.

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A247108 Complementary Aitken's array: triangle of numbers {a(n,k), n >= 0, 0<=k<=n} read by rows, defined by a(0,0)=1, a(n,0)=-a(n-1,n-1), a(n,k)=a(n,k-1)+a(n-1,k-1).

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A280470 Triangle A106534 with reversed rows.

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