A002411 -id:A002411 - cp's OEIS frontend

A334997 Array T read by ascending antidiagonals: T(n, k) = Sum_{d divides n} T(d, k-1) with T(n, 0) = 1.

1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 3, 3, 4, 1, 1, 2, 6, 4, 5, 1, 1, 4, 3, 10, 5, 6, 1, 1, 2, 9, 4, 15, 6, 7, 1, 1, 4, 3, 16, 5, 21, 7, 8, 1, 1, 3, 10, 4, 25, 6, 28, 8, 9, 1, 1, 4, 6, 20, 5, 36, 7, 36, 9, 10, 1, 1, 2, 9, 10, 35, 6, 49, 8, 45, 10, 11, 1, 1, 6, 3, 16, 15, 56, 7, 64, 9, 55, 11, 12, 1

Offset: 1

Stefano Spezia, May 19 2020

T(n, k) is called the generalized divisor function (see Beekman).

As an array with offset n=1, k=0, T(n,k) is the number of length-k chains of divisors of n. For example, the T(4,3) = 10 chains are: 111, 211, 221, 222, 411, 421, 422, 441, 442, 444. - Gus Wiseman, Aug 04 2022

			From _Gus Wiseman_, Aug 04 2022: (Start)
Array begins:
       k=0 k=1 k=2 k=3 k=4 k=5 k=6 k=7 k=8
  n=1:  1   1   1   1   1   1   1   1   1
  n=2:  1   2   3   4   5   6   7   8   9
  n=3:  1   2   3   4   5   6   7   8   9
  n=4:  1   3   6  10  15  21  28  36  45
  n=5:  1   2   3   4   5   6   7   8   9
  n=6:  1   4   9  16  25  36  49  64  81
  n=7:  1   2   3   4   5   6   7   8   9
  n=8:  1   4  10  20  35  56  84 120 165
The T(4,5) = 21 chains:
  (1,1,1,1,1)  (4,2,1,1,1)  (4,4,2,2,2)
  (2,1,1,1,1)  (4,2,2,1,1)  (4,4,4,1,1)
  (2,2,1,1,1)  (4,2,2,2,1)  (4,4,4,2,1)
  (2,2,2,1,1)  (4,2,2,2,2)  (4,4,4,2,2)
  (2,2,2,2,1)  (4,4,1,1,1)  (4,4,4,4,1)
  (2,2,2,2,2)  (4,4,2,1,1)  (4,4,4,4,2)
  (4,1,1,1,1)  (4,4,2,2,1)  (4,4,4,4,4)
The T(6,3) = 16 chains:
  (1,1,1)  (3,1,1)  (6,2,1)  (6,6,1)
  (2,1,1)  (3,3,1)  (6,2,2)  (6,6,2)
  (2,2,1)  (3,3,3)  (6,3,1)  (6,6,3)
  (2,2,2)  (6,1,1)  (6,3,3)  (6,6,6)
The triangular form T(n-k,k) gives the number of length k chains of divisors of n - k. It begins:
  1
  1  1
  1  2  1
  1  2  3  1
  1  3  3  4  1
  1  2  6  4  5  1
  1  4  3 10  5  6  1
  1  2  9  4 15  6  7  1
  1  4  3 16  5 21  7  8  1
  1  3 10  4 25  6 28  8  9  1
  1  4  6 20  5 36  7 36  9 10  1
  1  2  9 10 35  6 49  8 45 10 11  1
(End)

Richard Beekman, An Introduction to Number-Theoretic Combinatorics, Lulu Press 2017.

Stefano Spezia, First 150 antidiagonals of the array, flattened
Richard Beekman, A General Solution of the Ferris Wheel Problem.

Cf. A000217 (4th row), A000290 (6th row), A000292 (8th row), A000332 (16th row), A000389 (32nd row), A000537 (36th row), A000578 (30th row), A002411 (12th row), A002417 (24th row), A007318, A027800 (48th row), A335078, A335079.
Column k = 2 of the array is A007425.
Column k = 3 of the array is A007426.
Column k = 4 of the array is A061200.
The transpose of the array is A077592.
The subdiagonal n = k + 1 of the array is A163767.
The version counting all multisets of divisors (not just chains) is A343658.
The strict case is A343662 (row sums: A337256).
Diagonal n = k of the array is A343939.
Antidiagonal sums of the array (or row sums of the triangle) are A343940.
A067824(n) counts strict chains of divisors starting with n.
A074206(n) counts strict chains of divisors from n to 1.
A146291 counts divisors by Omega.
A251683(n,k) counts strict length k + 1 chains of divisors from n to 1.
A253249(n) counts nonempty chains of divisors of n.
A334996(n,k) counts strict length k chains of divisors from n to 1.
A337255(n,k) counts strict length k chains of divisors starting with n.
Cf. A000005, A018892, A051026, A062319, A066959, A176029, A327527, A343652, A343656.

T[n_,k_]:=If[n==1,1,Product[Binomial[Extract[Extract[FactorInteger[n],i],2]+k,k],{i,1,Length[FactorInteger[n]]}]]; Table[T[n-k,k],{n,1,13},{k,0,n-1}]//Flatten

T(n, k) = if (k==0, 1, sumdiv(n, d, T(d, k-1)));
matrix(10, 10, n, k, T(n, k-1)) \\ to see the array for n>=1, k >=0; \\ Michel Marcus, May 20 2020

T(n, k) = Sum_{d divides n} T(d, k-1) with T(n, 0) = 1 (see Theorem 3 in Beekman's article).
T(i*j, k) = T(i, k)*T(j, k) if i and j are coprime positive integers (see Lemma 1 in Beekman's article).
T(p^m, k) = binomial(m+k, k) for every prime p (see Lemma 2 in Beekman's article).

Duplicate term removed by Stefano Spezia, Jun 03 2020

A001499 Number of n X n matrices with exactly 2 1's in each row and column, other entries 0.

Original entry on oeis.org

1, 0, 1, 6, 90, 2040, 67950, 3110940, 187530840, 14398171200, 1371785398200, 158815387962000, 21959547410077200, 3574340599104475200, 676508133623135814000, 147320988741542099484000, 36574751938491748341360000, 10268902998771351157327104000

Offset: 0

N. J. A. Sloane

Or, number of labeled 2-regular relations of order n.

Also number of ways to arrange 2n rooks on an n X n chessboard, with no more than 2 rooks in each row and column (no 3 in a line). - Vaclav Kotesovec, Aug 03 2013

R. Bricard, L'Intermédiaire des Mathématiciens, 8 (1901), 312-313.
L. Comtet, Advanced Combinatorics, Reidel, 1974, Sect. 6.3 Multipermutations, pp. 235-236, P(n,2), bipermutations.
L. Erlebach and O. Ruehr, Problem 79-5, SIAM Review. Solution by D. E. Knuth. Reprinted in Problems in Applied Mathematics, ed. M. Klamkin, SIAM, 1990, p. 350.
Shanzhen Gao and Kenneth Matheis, Closed formulas and integer sequences arising from the enumeration of (0,1)-matrices with row sum two and some constant column sums. In Proceedings of the Forty-First Southeastern International Conference on Combinatorics, Graph Theory and Computing. Congr. Numer. 202 (2010), 45-53.
J. T. Lewis, Maximal L-free subsets of a squarefree array, Congressus Numerantium, 141 (1999), 151-155.
R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1976.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Cor. 5.5.11 (b).
M. L. Stein and P. R. Stein, Enumeration of Stochastic Matrices with Integer Elements. Report LA-4434, Los Alamos Scientific Laboratory of the University of California, Los Alamos, NM, Jun 1970.
J. H. van Lint and R. M. Wilson, A Course in Combinatorics (Cambridge University Press, Cambridge, 1992), pp. 152-153. [The second edition is said to be a better reference.]

Alois P. Heinz, Table of n, a(n) for n = 0..200 (terms n = 0..48 from R. W. Robinson)
H. Anand, V. C. Dumir and H. Gupta, A combinatorial distribution problem, Duke Math. J., 33 (1996), 757-769.
Paul Barry, On the Connection Coefficients of the Chebyshev-Boubaker polynomials, The Scientific World Journal, Volume 2013 (2013), Article ID 657806.
L. Carlitz, Enumeration of symmetric arrays, Duke Math. J., Vol. 33 (1966), 771-782.
Sally Cockburn and Joshua Lesperance, Deranged Socks, Mathematics Magazine, Vol. 86, no. 2, April 2013, pp. 97-109.
L. Erlebach and O. Ruehr, Problem 79-5, SIAM Review, Vol. 21, No. 1 (Jan., 1979), p. 140. Solution by D. E. Knuth, SIAM Review, Vol. 22, No. 1 (Jan., 1980), pp. 101-102.
M. E. Kuczma, 0-1-Matrices with Line-Sums Equal to 2, Am. Math. Month. 99 (1992) 959-961, E3419.
Rui-Li Liu and Feng-Zhen Zhao, New Sufficient Conditions for Log-Balancedness, With Applications to Combinatorial Sequences, J. Int. Seq., Vol. 21 (2018), Article 18.5.7.
M. L. Stein and P. R. Stein, Enumeration of Stochastic Matrices with Integer Elements, Report LA-4434, Los Alamos Scientific Laboratory of the University of California, Los Alamos, NM, Jun 1970. [Annotated scanned copy]
Zhonghua Tan, Shanzhen Gao, and H. Niederhausen, Enumeration of (0,1) matrices with constant row and column sums, Appl. Math. Chin. Univ. 21 (4) (2006) 479-486.
Bo-Ying Wang and Fuzhen Zhang, On the precise number of (0,1)-matrices in A(R,S), Discrete Math. 187 (1998), no. 1-3, 211--220. MR1630720 (99f:05010). - From _N. J. A. Sloane_, Jun 07 2012
Index entries for sequences related to binary matrices

Cf. A000681, A053871, A123544 (connected relations), A000986 (symmetric matrices), A007107 (traceless matrices).
Cf. A000290, A002411, A005408.
Cf. A001501. Column 2 of A008300. Row sums of A284989.

a001499 n = a001499_list !! n
a001499_list = 1 : 0 : 1 : zipWith (*) (drop 2 a002411_list)
   (zipWith (+) (zipWith (*) [3, 5 ..] $ tail a001499_list)
                (zipWith (*) (tail a000290_list) a001499_list))
-- Reinhard Zumkeller, Jun 02 2013

a[n_] := (n-1)*n!*Gamma[n-1/2]*Hypergeometric1F1[2-n, 3/2-n, -1/2]/Sqrt[Pi]; Table[a[n], {n, 0, 17}] (* Jean-François Alcover, Oct 06 2011, after first formula *)

a(n)=if(n<2,n==0,(n^2-n)*(a(n-1)+(n-1)/2*a(n-2)))

seq(n)={Vec(serlaplace(serlaplace(exp(-x/2 + O(x*x^n))/sqrt(1-x + O(x*x^n)))))}; \\ Andrew Howroyd, Sep 09 2018

a(n) = (n! (n-1) Gamma(n-1/2) / Gamma(1/2) ) * 1F1[2-n; 3/2-n; -1/2] [Erlebach and Ruehr]. This representation is exact, asymptotic and convergent.
D-finite with recurrence 2*a(n) -2*n*(n-1)*a(n-1) -n*(n-1)^2*a(n-2)=0.
a(n) ~ 2 sqrt(Pi) n^(2n + 1/2) e^(-2n - 1/2) [Knuth]
a(n) = (1/2)*n*(n-1)^2 * ( (2*n-3)*a(n-2) + (n-2)^2*a(n-3) ) (from Anand et al.)
Sum_{n >= 0} a(n)*x^n/(n!)^2 = exp(-x/2)/sqrt(1-x); a(n) = n(n-1)/2 [ 2 a(n-1) + (n-1) a(n-2) ] (Bricard)
b_n = a_n/n! satisfies b_n = (n-1)(b_{n-1} + b_{n-2}/2); e.g.f. for {b_n} and for derangements (A000166) are related by D(x) = B(x)^2.
Limit_(n->infinity) sqrt(n)*a(n)/(n!)^2 = A096411 [Kuczma]. - R. J. Mathar, Sep 21 2007
a(n) = 4^(-n) * n!^2 * Sum_{i=0..n} (-2)^i * (2*n - 2*i)! / (i!*(n-i)!^2). - Shanzhen Gao, Feb 15 2010

A077592 Table by antidiagonals of tau_k(n), the k-th Piltz function (see A007425), or n-th term of the sequence resulting from applying the inverse Möbius transform (k-1) times to the all-ones sequence.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 4, 3, 3, 1, 1, 5, 4, 6, 2, 1, 1, 6, 5, 10, 3, 4, 1, 1, 7, 6, 15, 4, 9, 2, 1, 1, 8, 7, 21, 5, 16, 3, 4, 1, 1, 9, 8, 28, 6, 25, 4, 10, 3, 1, 1, 10, 9, 36, 7, 36, 5, 20, 6, 4, 1, 1, 11, 10, 45, 8, 49, 6, 35, 10, 9, 2, 1, 1, 12, 11, 55, 9, 64, 7, 56, 15, 16, 3, 6, 1

Offset: 1

Henry Bottomley, Nov 08 2002

As an array with offset n=0, k=1, also the number of length n chains of divisors of k. - Gus Wiseman, Aug 04 2022

			T(6,3) = 9 because we have: 1*1*6, 1*2*3, 1*3*2, 1*6*1, 2*1*3, 2*3*1, 3*1*2, 3*2*1, 6*1*1. - _Geoffrey Critzer_, Feb 16 2015
From _Gus Wiseman_, May 03 2021: (Start)
Array begins:
       k=1 k=2 k=3 k=4 k=5 k=6 k=7 k=8
  n=0:  1   1   1   1   1   1   1   1
  n=1:  1   2   2   3   2   4   2   4
  n=2:  1   3   3   6   3   9   3  10
  n=3:  1   4   4  10   4  16   4  20
  n=4:  1   5   5  15   5  25   5  35
  n=5:  1   6   6  21   6  36   6  56
  n=6:  1   7   7  28   7  49   7  84
  n=7:  1   8   8  36   8  64   8 120
  n=8:  1   9   9  45   9  81   9 165
The triangular form T(n,k) = A(n-k,k) gives the number of length n - k chains of divisors of k. It begins:
  1
  1  1
  1  2  1
  1  3  2  1
  1  4  3  3  1
  1  5  4  6  2  1
  1  6  5 10  3  4  1
  1  7  6 15  4  9  2  1
  1  8  7 21  5 16  3  4  1
  1  9  8 28  6 25  4 10  3  1
  1 10  9 36  7 36  5 20  6  4  1
  1 11 10 45  8 49  6 35 10  9  2  1
(End)

Alois P. Heinz, Antidiagonals n = 1..141, flattened
Adolf Piltz, Ueber das Gesetz, nach welchem die mittlere Darstellbarkeit der natürlichen Zahlen als Produkte einer gegebenen Anzahl Faktoren mit der Grösse der Zahlen wächst, Doctoral Dissertation, Friedrich-Wilhelms-Universität zu Berlin, 1881; the k-th Piltz function tau_k(n) is denoted by phi(n,k) and its recurrence and Dirichlet series appear on p. 6.
Wikipedia, Adolf Piltz.

Rows include: A000012, A000005, A034695, A111217, A111218, A111219, A111220, A111221, A111306.
Columns include (with multiplicity and some offsets) A000012, A000027, A000027, A000217, A000027, A000290, A000027, A000292, A000217, A000290, A000027, A002411, A000027, A000290, A000290, A000332 etc.
Cf. A077593.
Row n = 2 of the array is A007425.
Row n = 3 of the array is A007426.
Row n = 4 of the array is A061200.
The diagonal n = k of the array (central column of the triangle) is A163767.
The transpose of the array is A334997.
Diagonal n = k of the array is A343939.
Antidiagonal sums of the array (or row sums of the triangle) are A343940.
A067824(n) counts strict chains of divisors starting with n.
A074206(n) counts strict chains of divisors from n to 1.
A146291(n,k) counts divisors of n with k prime factors (with multiplicity).
A251683(n,k) counts strict length k + 1 chains of divisors from n to 1.
A253249(n) counts nonempty chains of divisors of n.
A334996(n,k) counts strict length k chains of divisors from n to 1.
A337255(n,k) counts strict length k chains of divisors starting with n.
Cf. A018892, A051026, A062319, A143773, A176029, A327527, A337256, A343656, A343658, A343662.

with(numtheory):
A:= proc(n,k) option remember; `if`(k=1, 1,
      add(A(d, k-1), d=divisors(n)))
    end:
seq(seq(A(n, 1+d-n), n=1..d), d=1..14);  # Alois P. Heinz, Feb 25 2015

tau[n_, 1] = 1; tau[n_, k_] := tau[n, k] = Plus @@ (tau[ #, k - 1] & /@ Divisors[n]); Table[tau[n - k + 1, k], {n, 14}, {k, n, 1, -1}] // Flatten (* Robert G. Wilson v *)
tau[1, k_] := 1; tau[n_, k_] := Times @@ (Binomial[Last[#] + k - 1, k - 1] & /@ FactorInteger[n]); Table[tau[k, n - k + 1], {n, 1, 13}, {k, 1, n}] // Flatten (* Amiram Eldar, Sep 13 2020 *)
Table[Length[Select[Tuples[Divisors[k],n-k],And@@Divisible@@@Partition[#,2,1]&]],{n,12},{k,1,n}] (* TRIANGLE, Gus Wiseman, May 03 2021 *)
Table[Length[Select[Tuples[Divisors[k],n-1],And@@Divisible@@@Partition[#,2,1]&]],{n,6},{k,6}] (* ARRAY, Gus Wiseman, May 03 2021 *)

If n = Product_i p_i^e_i, then T(n,k) = Product_i C(k+e_i-1, e_i). T(n,k) = Sum_d{d|n} T(n-1,d) = A077593(n,k) - A077593(n-1,k).
Columns are multiplicative.
Dirichlet g.f. for column k: Zeta(s)^k. - Geoffrey Critzer, Feb 16 2015
A(n,k) = A334997(k,n). - Gus Wiseman, Aug 04 2022

Typo in formula fixed by Geoffrey Critzer, Feb 16 2015

A094727 Triangle read by rows: T(n,k) = n + k, 0 <= k < n, n >= 1.

Original entry on oeis.org

1, 2, 3, 3, 4, 5, 4, 5, 6, 7, 5, 6, 7, 8, 9, 6, 7, 8, 9, 10, 11, 7, 8, 9, 10, 11, 12, 13, 8, 9, 10, 11, 12, 13, 14, 15, 9, 10, 11, 12, 13, 14, 15, 16, 17, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23

Offset: 1

Reinhard Zumkeller, May 24 2004

All numbers m occur ceiling(m/2) times, see A004526.

The LCM of the n-th row is A076100. - Michel Marcus, Mar 18 2018

			Triangle begins:
  1;
  2,  3;
  3,  4,  5;
  4,  5,  6,  7;
  5,  6,  7,  8,  9;
  6,  7,  8,  9, 10, 11;
  7,  8,  9, 10, 11, 12, 13;
  8,  9, 10, 11, 12, 13, 14, 15;
  9, 10, 11, 12, 13, 14, 15, 16, 17;
  ... - _Philippe Deléham_, Mar 30 2013

Reinhard Zumkeller, Rows n = 1..150 of triangle, flattened
Bruno Berselli, Illustration of the initial terms.
László Németh, On the Binomial Interpolated Triangles, Journal of Integer Sequences, Vol. 20 (2017), Article 17.7.8.
Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions, arXiv:1212.2732 [math.CO], 2012.

Cf. A002024, A002260, A003056, A004526, A004736, A005408, A005843.
Cf. A016777, A016813, A016861, A037213, A076100, A078112, A093005.
Cf. A094728, A214604, A214661, A319556.
Cf. A128076 (rows reversed).

a094727 n k = n + k
a094727_row n = a094727_tabl !! (n-1)
a094727_tabl = iterate (\row@(h:_) -> (h + 1) : map (+ 2) row) [1]
-- Reinhard Zumkeller, Jul 22 2012

z:=12; &cat[ [m+n-1: m in [1..n] ]: n in [1..z] ];

Table[n + Range[0, n-1], {n, 12}]//Flatten (* Michael De Vlieger, Dec 16 2016 *)

from math import isqrt
def A094727(n): return ((a:=(m:=isqrt(k:=n<<1))+(k>m*(m+1)))*(3-a)>>1)+n-1 # Chai Wah Wu, Jun 19 2025

flatten([[n+k for k in range(n)] for n in range(1,16)]) # G. C. Greubel, Mar 10 2024

T(n+1, k) = T(n, k) + 1 = T(n, k+1); T(n+1, k+1) = T(n, k) + 2.
T(n, n - A005843(k)) = A005843(n-k) for 0 <= k <= n/2.
T(n, n - A005408(k)) = A005408(n-k) for 0 <= k < n/2.
T(A005408(n), n) = A016777(n), n >= 0.
Sum_{k=1..n} T(n, k) = A000326(n) (row sums).
T(n, k) = A002024(n,k) + A002260(n,k) - 1. - Reinhard Zumkeller, Apr 27 2006
As a sequence rather than as a table: If m = floor((sqrt(8n-7)+1)/2), a(n) = n - m*(m-3)/2 - 1. - Carl R. White, Jul 30 2009
T(n, k) = n+k-1, n >= k >= 1. - Vincenzo Librandi, Nov 23 2009 [corrected by Klaus Brockhaus, Nov 23 2009]
T(n,k) = A037213((A214604(n,k) + A214661(n,k)) / 2). - Reinhard Zumkeller, Jul 25 2012
From Boris Putievskiy, Jan 16 2013: (Start)
a(n) = A002260(n) + A003056(n).
a(n) = i+t, where i=n-t*(t+1)/2, t=floor((-1+sqrt(8*n-7))/2). (End)
From G. C. Greubel, Mar 10 2024: (Start)
T(3*n-3, n) = A016813(n-1).
T(4*n-4, n) = A016861(n-1).
Sum_{k=0..n-1} (-1)^k*T(n, k) = A319556(n).
Sum_{k=0..floor((n-1)/2)} T(n-k, k) = A093005(n).
Sum_{k=0..floor((n-1)/2)} (-1)^k*T(n-k, k) = A078112(n-1).
Sum_{j=1..n} (Sum_{k=0..n-1} T(j, k)) = A002411(n) (sum of n rows). (End)

A093560 (3,1) Pascal triangle.

Original entry on oeis.org

1, 3, 1, 3, 4, 1, 3, 7, 5, 1, 3, 10, 12, 6, 1, 3, 13, 22, 18, 7, 1, 3, 16, 35, 40, 25, 8, 1, 3, 19, 51, 75, 65, 33, 9, 1, 3, 22, 70, 126, 140, 98, 42, 10, 1, 3, 25, 92, 196, 266, 238, 140, 52, 11, 1, 3, 28, 117, 288, 462, 504, 378, 192, 63, 12, 1, 3, 31, 145, 405, 750, 966, 882, 570, 255, 75, 13, 1

Offset: 0

Wolfdieter Lang, Apr 22 2004

The array F(3;n,m) gives in the columns m >= 1 the figurate numbers based on A016777, including the pentagonal numbers A000326 (see the W. Lang link).
This is the third member, d=3, in the family of triangles of figurate numbers, called (d,1) Pascal triangles: A007318 (Pascal (d=1), A029653 (d=2).
This is an example of a Riordan triangle (see A053121 for a comment and the 1991 Shapiro et al. reference on the Riordan group) with o.g.f. of column no. m of the type g(x)*(x*f(x))^m with f(0)=1. Therefore the o.g.f. for the row polynomials p(n,x):=Sum_{m=0..n} a(n,m)*x^m is G(z,x)=g(z)/(1-x*z*f(z)). Here: g(x)=(1+2*x)/(1-x), f(x)=1/(1-x), hence G(z,x)=(1+2*z)/(1-(1+x)*z).
The SW-NE diagonals give the Lucas numbers A000032: L(n) = Sum_{k=0..ceiling((n-1)/2)} a(n-1-k,k), n >= 1, with L(0)=2. Observation by Paul Barry, Apr 29 2004. Proof via recursion relations and comparison of inputs.
Triangle T(n,k), read by rows, given by [3,-2,0,0,0,0,0,0,...] DELTA [1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Sep 17 2009
For a closed-form formula for generalized Pascal's triangle see A228576. - Boris Putievskiy, Sep 09 2013
From Wolfdieter Lang, Jan 09 2015: (Start)
The signed lower triangular matrix (-1)^(n-1)*a(n,m) is the inverse of the Riordan matrix A106516; that is Riordan ((1-2*x)/(1+x),x/(1+x)).
See the Peter Bala comment from Dec 23 2014 in A106516 for general Riordan triangles of the type (g(x), x/(1-x)): exp(x)*r(n,x) = d(n,x) with the e.g.f. r(n,x) of row n and the e.g.f. of diagonal n.
Similarly, for general Riordan triangles of the type (g(x), x/(1+x)): exp(x)*r(n,-x) = d(n,x). (End)
The n-th row polynomial is (3 + x)*(1 + x)^(n-1) for n >= 1. More generally, the n-th row polynomial of the Riordan array ( (1-a*x)/(1-b*x), x/(1-b*x) ) is (b - a + x)*(b + x)^(n-1) for n >= 1. - Peter Bala, Mar 02 2018
Binomial(n-2,k)+2*Binomial(n-3,k) is also the number of permutations avoiding both 123 and 132 with k double descents, i.e., positions with w[i]>w[i+1]>w[i+2]. - Lara Pudwell, Dec 19 2018

			Triangle begins
  1,
  3,  1,
  3,  4,  1,
  3,  7,  5,   1,
  3, 10, 12,   6,   1,
  3, 13, 22,  18,   7,   1,
  3, 16, 35,  40,  25,   8,   1,
  3, 19, 51,  75,  65,  33,   9,  1,
  3, 22, 70, 126, 140,  98,  42, 10,  1,
  3, 25, 92, 196, 266, 238, 140, 52, 11, 1,

Kurt Hawlitschek, Johann Faulhaber 1580-1635, Veroeffentlichung der Stadtbibliothek Ulm, Band 18, Ulm, Germany, 1995, Ch. 2.1.4. Figurierte Zahlen.
Ivo Schneider, Johannes Faulhaber 1580-1635, Birkhäuser, Basel, Boston, Berlin, 1993, ch.5, pp. 109-122.

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
Peter Bala, A note on the diagonals of a proper Riordan Array
M. Bukata, R. Kulwicki, N. Lewandowski, L. Pudwell, J. Roth, and T. Wheeland, Distributions of Statistics over Pattern-Avoiding Permutations, arXiv preprint arXiv:1812.07112 [math.CO], 2018.
W. Lang, First 10 rows and array of figurate numbers .

Cf. Column sequences for m=1..9: A016777, A000326 (pentagonal), A002411, A001296, A051836, A051923, A050494, A053367, A053310;

A007318 (Pascal's triangle), A029653 ((2,1) Pascal triangle), A093561 ((4,1) Pascal triangle), A228196, A228576.

Concatenation([1],Flat(List([1..11],n->List([0..n],k->Binomial(n,k)+2*Binomial(n-1,k))))); # Muniru A Asiru, Dec 20 2018

a093560 n k = a093560_tabl !! n !! k
a093560_row n = a093560_tabl !! n
a093560_tabl = [1] : iterate
               (\row -> zipWith (+) ([0] ++ row) (row ++ [0])) [3, 1]
-- Reinhard Zumkeller, Aug 31 2014

from math import comb, isqrt
def A093560(n): return comb(r:=(m:=isqrt(k:=n+1<<1))-(k<=m*(m+1)),a:=n-comb(r+1,2))*(r+(r-a<<1))//r if n else 1 # Chai Wah Wu, Nov 12 2024

a(n, m)=F(3;n-m, m) for 0<= m <= n, otherwise 0, with F(3;0, 0)=1, F(3;n, 0)=3 if n>=1 and F(3;n, m):=(3*n+m)*binomial(n+m-1, m-1)/m if m>=1.
G.f. column m (without leading zeros): (1+2*x)/(1-x)^(m+1), m>=0.
Recursion: a(n, m)=0 if m>n, a(0, 0)= 1; a(n, 0)=3 if n>=1; a(n, m)= a(n-1, m) + a(n-1, m-1).
T(n, k) = C(n, k) + 2*C(n-1, k). - Philippe Deléham, Aug 28 2005
Equals M * A007318, where M = an infinite triangular matrix with all 1's in the main diagonal and all 2's in the subdiagonal. - Gary W. Adamson, Dec 01 2007
Sum_{k=0..n} T(n,k) = A151821(n+1). - Philippe Deléham, Sep 17 2009
exp(x) * e.g.f. for row n = e.g.f. for diagonal n. For example, for n = 3 we have exp(x)*(3 + 7*x + 5*x^2/2! + x^3/3!) = 3 + 10*x + 22*x^2/2! + 40*x^3/3! + 65*x^4/4! + .... The same property holds more generally for Riordan arrays of the form ( f(x), x/(1 - x) ). - Peter Bala, Dec 22 2014
G.f.: (-1-2*x)/(-1+x+x*y). - R. J. Mathar, Aug 11 2015

Incorrect connection with A046055 deleted by N. J. A. Sloane, Jul 08 2009

A139601 Square array of polygonal numbers read by ascending antidiagonals: T(n, k) = (n + 1)(k - 1)k/2 + k.

Original entry on oeis.org

0, 0, 1, 0, 1, 3, 0, 1, 4, 6, 0, 1, 5, 9, 10, 0, 1, 6, 12, 16, 15, 0, 1, 7, 15, 22, 25, 21, 0, 1, 8, 18, 28, 35, 36, 28, 0, 1, 9, 21, 34, 45, 51, 49, 36, 0, 1, 10, 24, 40, 55, 66, 70, 64, 45, 0, 1, 11, 27, 46, 65, 81, 91, 92, 81, 55, 0, 1, 12, 30, 52, 75, 96, 112, 120, 117, 100, 66

Offset: 0

Omar E. Pol, Apr 27 2008

A general formula for polygonal numbers is P(n,k) = (n-2)(k-1)k/2 + k, where P(n,k) is the k-th n-gonal number. - Omar E. Pol, Dec 21 2008

			The square array of polygonal numbers begins:
========================================================
Triangulars .. A000217: 0, 1,  3,  6, 10,  15,  21,  28,
Squares ...... A000290: 0, 1,  4,  9, 16,  25,  36,  49,
Pentagonals .. A000326: 0, 1,  5, 12, 22,  35,  51,  70,
Hexagonals ... A000384: 0, 1,  6, 15, 28,  45,  66,  91,
Heptagonals .. A000566: 0, 1,  7, 18, 34,  55,  81, 112,
Octagonals ... A000567: 0, 1,  8, 21, 40,  65,  96, 133,
9-gonals ..... A001106: 0, 1,  9, 24, 46,  75, 111, 154,
10-gonals .... A001107: 0, 1, 10, 27, 52,  85, 126, 175,
11-gonals .... A051682: 0, 1, 11, 30, 58,  95, 141, 196,
12-gonals .... A051624: 0, 1, 12, 33, 64, 105, 156, 217,
And so on ..............................................
========================================================

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Peter Luschny, Figurate number — a very short introduction. With plots from Stefan Friedrich Birkner.
Omar E. Pol, Polygonal numbers, An alternative illustration of initial terms.

Sequences of m-gonal numbers: A000217 (m=3), A000290 (m=4), A000326 (m=5), A000384 (m=6), A000566 (m=7), A000567 (m=8), A001106 (m=9), A001107 (m=10), A051682 (m=11), A051624 (m=12), A051865 (m=13), A051866 (m=14), A051867 (m=15), A051868 (m=16), A051869 (m=17), A051870 (m=18), A051871 (m=19), A051872 (m=20), A051873 (m=21), A051874 (m=22), A051875 (m=23), A051876 (m=24), A255184 (m=25), A255185 (m=26), A255186 (m=27), A161935 (m=28), A255187 (m=29), A254474 (m=30).
Cf. A000007, A000012, A000027, A002411, A006000, A006003, A006522.
Cf. A008585, A016957, A017329, A086270, A117142, A139606, A139607.
Cf. A139608, A139609, A139610, A139611, A139612, A139613, A139614.
Cf. A139615, A139616, A057145, A086271, A139600, A139617, A139618.
Cf. A139619, A139620.

T:= func< n,k | k*((n+1)*(k-1) +2)/2 >;
A139601:= func< n,k | T(n-k, k) >;
[A139601(n,k): k in  [0..n], n in [0..12]]; // G. C. Greubel, Jul 12 2024

T[n_, k_] := (n + 1)*(k - 1)*k/2 + k; Table[ T[n - k, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Robert G. Wilson v, Jul 12 2009 *)

def T(n,k): return k*((n+1)*(k-1)+2)/2
def A139601(n,k): return T(n-k, k)
flatten([[A139601(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jul 12 2024

T(n,k) = A086270(n,k), k>0. - R. J. Mathar, Aug 06 2008
T(n,k) = (n+1)*(k-1)*k/2 +k, n>=0, k>=0. - Omar E. Pol, Jan 07 2009
From G. C. Greubel, Jul 12 2024: (Start)
t(n, k) = (k/2)*( (k-1)*(n-k+1) + 2), where t(n,k) is this array read by rising antidiagonals.
t(2*n, n) = A006003(n).
t(2*n+1, n) = A002411(n).
t(2*n-1, n) = A006000(n-1).
Sum_{k=0..n} t(n, k) = A006522(n+2).
Sum_{k=0..n} (-1)^k*t(n, k) = (-1)^n * A117142(n).
Sum_{k=0..n} t(n-k, k) = (2*n^4 + 34*n^2 + 48*n - 15 + 3*(-1)^n*(2*n^2 + 16*n + 5))/384. (End)

A033994 a(n) = n(n+1)(5*n+1)/6.

Original entry on oeis.org

2, 11, 32, 70, 130, 217, 336, 492, 690, 935, 1232, 1586, 2002, 2485, 3040, 3672, 4386, 5187, 6080, 7070, 8162, 9361, 10672, 12100, 13650, 15327, 17136, 19082, 21170, 23405, 25792, 28336, 31042, 33915, 36960, 40182, 43586, 47177, 50960, 54940

Offset: 1

Barry E. Williams, Dec 16 1999

Partial sums of A005476.

a(n) is the dot product of the vectors of the first n positive integers and the next n integers. - Michel Marcus, Sep 02 2020

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A005476, A016873, A000330, A132124, A132112, A050409.

Cf. A000292, A001622, A002411, A002412.

a:=List([1..40],n->n*(n+1)*(5*n+1)/6);; Print(a); # Muniru A Asiru, Jan 01 2019

[n*(n+1)*(5*n+1)/6 : n in [1..40]]; // Vincenzo Librandi, Jan 01 2019

[n*(n+1)*(5*n+1)/6$n=1..40]; # Muniru A Asiru, Jan 01 2019

Table[Range[x].Range[x+1,2x],{x,40}] (* or *) LinearRecurrence[{4,-6,4,-1},{2,11,32,70},40] (* Harvey P. Dale, Jun 01 2018 *)

PARI
```
a(n) = n*(n+1)*(5*n+1)/6;
```

G.f.: x*(2+3*x)/(1-x)^4.
a(n) = A132121(n,1). - Reinhard Zumkeller, Aug 12 2007
a(n) = A000292(n) + A002412(n) = A000330(n) + A002411(n). - Omar E. Pol, Jan 11 2013
a(n) = Sum_{i=1..n} Sum_{j=1..n} i+min(i,j). - Enrique Pérez Herrero, Jan 15 2013
a(n) = Sum_{i=1..n} i*(n+i). - Charlie Marion, Apr 10 2013
Sum_{n>=1} 1/a(n) = 36 - 3*Pi*5^(3/4)*phi^(3/2)/4 - 15*sqrt(5)*log(phi)/4 - 75*log(5)/8 = 0.66131826232008423794478..., where phi = A001622 = (1 + sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jun 01 2018
E.g.f.: exp(x)*x*(12 + 21*x + 5*x^2)/6. - Stefano Spezia, Feb 21 2024

More terms from James Sellers, Jan 19 2000

A075362 Triangle read by rows with the n-th row containing the first n multiples of n.

Original entry on oeis.org

1, 2, 4, 3, 6, 9, 4, 8, 12, 16, 5, 10, 15, 20, 25, 6, 12, 18, 24, 30, 36, 7, 14, 21, 28, 35, 42, 49, 8, 16, 24, 32, 40, 48, 56, 64, 9, 18, 27, 36, 45, 54, 63, 72, 81, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 121, 12, 24, 36, 48, 60, 72, 84

Offset: 1

Amarnath Murthy, Sep 20 2002

(Conjecture) Let N=2*n and k=1,...,n. Let A_{N,0}, A_{N,1}, ..., A_{N,n-1} be the n X n unit-primitive matrices (see [Jeffery]) associated with N. Define the Chebyshev polynomials of the second kind by the recurrence U_0(x)=1, U_1(x)=2*x and U_r(x)=2*x*U_(r-1)(x)-U_(r-2)(x) (r>1). Define the column vectors V_(k-1)=(U_(k-1)(cos(Pi/N)), U_(k-1)(cos(3*Pi/N)), ..., U_(k-1)(cos((2*n-1)*Pi/N)))^T, where B^T denotes the transpose of matrix B. Let S_N=[V_0,V_1,...,V_(n-1)] be the n X n matrix formed by taking the components of vector V_(k-1) as the entries in column k-1 (V_(k-1) gives the ordered spectrum of A_{N,k-1}). Let X_N=[S_N]^T*S_N, and let [X_N](i,j) denote the entry in row i and column j of X_N, i,j in {0,...,n-1}. Then also T(n,k)=[X_N](k-1,k-1); that is, row n of the triangle is given by the main diagonal entries of X_N. Hence T(n,k) is the sum of squares T(n,k) = sum[m=1,...,n (U_(k-1)(cos((2*m-1)*Pi/N)))^2]=[V_(k-1)]^T*V_(k-1). - L. Edson Jeffery, Jan 20 2012
Conjecture that antidiagonal sums are A023855. - L. Edson Jeffery, Jan 20 2012
Viewed as a sequence of rows, consider the subsequences (of rows) that contain every positive integer. The lexicographically latest of these subsequences consists of the rows with row numbers in A066680 U {1}; this is the only one that contains its own row numbers only once. - Peter Munn, Dec 04 2019

			Triangle begins:
  1;
  2,  4;
  3,  6,  9;
  4,  8, 12, 16;
  5, 10, 15, 20, 25;
  6, 12, 18, 24, 30, 36;

Reinhard Zumkeller, Rows n = 1..150 of triangle, flattened
L. E. Jeffery, Unit-primitive matrices

A002411 gives the sum of the n-th row. A141419 is similarly derived.
Cf. A066680, A223544.
Cf. A003991 (square multiplication table).
Main diagonal gives A000290.

a075362 n k = a075362_tabl !! (n-1) !! (k-1)
a075362_row n = a075362_tabl !! (n-1)
a075362_tabl = zipWith (zipWith (*)) a002260_tabl a002024_tabl
-- Reinhard Zumkeller, Nov 11 2012, Oct 04 2012

T(n,k):=piecewise(k<=n,sum(i*binomial(k,i)*binomial(n+1-k,n-i),i=1..k),k>n,0) # Mircea Merca, Apr 11 2012

Table[NestList[n+#&,n,n-1],{n,15}]//Flatten (* Harvey P. Dale, Jun 14 2022 *)

T(n,k) = n*k, 1 <= k <= n. - Reinhard Zumkeller, Mar 07 2010
T(n,k) = A050873(n,k)*A051173(n,k), 1 <= k <= n. - Reinhard Zumkeller, Apr 25 2011
T(n,k) = Sum_{i=1..k} i*binomial(k,i)*binomial(n+1-k,n-i), 1 <= k <= n. - Mircea Merca, Apr 11 2012
T(n,k) = A002260(n,k)*A002024(n,k) = (A215630(n,k)-A215631(n,k))/2, 1 <= k <= n. - Reinhard Zumkeller, Nov 11 2012
a(n) = A223544(n) - 1; a(n) = i*(t+1), where i = n - t*(t+1)/2, t = floor((-1 + sqrt(8*n-7))/2). - Boris Putievskiy, Jul 24 2013

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 20 2003

A088459 Triangle read by rows: T(n,k) represents the number of lozenge tilings of an (n,1,n)-hexagon which include the non-vertical tile above the main diagonal starting in position k+1.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 1, 3, 6, 6, 3, 1, 1, 4, 12, 18, 18, 12, 4, 1, 1, 5, 20, 40, 60, 60, 40, 20, 5, 1, 1, 6, 30, 75, 150, 200, 200, 150, 75, 30, 6, 1, 1, 7, 42, 126, 315, 525, 700, 700, 525, 315, 126, 42, 7, 1, 1, 8, 56, 196, 588, 1176, 1960, 2450, 2450, 1960, 1176, 588, 196, 56, 8, 1

Offset: 1

Christopher Hanusa (chanusa(AT)washington.edu), Nov 14 2003

Rows are of length 2, 4, 6, 8, 10, 12, ...
T(n,k)= number of symmetric Dyck paths of length 4n and having k peaks. Example: T(2,3)=2 because we have UU*DU*DU*DD and U*DUU*DDU*D, where U=(1,1), D=(1,-1) and * shows the peaks. - Emeric Deutsch, Feb 22 2004
T(n,k) is also the number of nodes at distance k from a specified node in the n-odd graph for k in 1..n-1. - Eric W. Weisstein, Mar 23 2018
T(n,k) seems to be the k-th Lie-Betti number of the star graph on n vertices. See A360571 for additional information and references. - Samuel J. Bevins, Feb 12 2023

			For example, the number of tilings of a 4,1,4 hexagon which includes the non-vertical tile above the main diagonal starting in position 3 is T(4,2)=12.
Triangle T(n, k) begins:
[1] 1,1,
[2] 1,2, 2,  1,
[3] 1,3, 6,  6,   3,   1,
[4] 1,4,12, 18,  18,  12,   4,   1,
[5] 1,5,20, 40,  60,  60,  40,  20,   5,   1,
[6] 1,6,30, 75, 150, 200, 200, 150,  75,  30,   6,  1,
[7] 1,7,42,126, 315, 525, 700, 700, 525, 315, 126, 42,    7,   1,
[8] 1,8,56,196, 588,1176,1960,2450,2450,1960,1176,588,  196,  56,  8, 1,
[9] 1,9,72,288,1008,2352,4704,7056,8820,8820,7056,4704,2352,1008,288,72,9,1

Columns 0-5 are sequences A000012, A000027, A002378, A002411, A006011 and A004302.

Cf. A000984 (row sums).

A088459 := proc(n,k)
    binomial(n,ceil(k/2))*binomial(n-1,floor(k/2)) ;
end proc:
seq(seq(A088459(n,k),k=0..2*n-1),n=1..10) ; # R. J. Mathar, Apr 02 2017

Table[Binomial[n, Ceiling[k/2]] Binomial[n - 1, Floor[k/2]], {n, 10}, {k, 0, 2 n - 1}] // Flatten (* Eric W. Weisstein, Mar 23 2018 *)

T(n, k) = binomial(n, ceiling(k/2))* binomial(n-1, floor(k/2)), n>0 and k=0 to 2n-1.

Edited and extended by Ray Chandler, Nov 17 2003

A285824 Number T(n,k) of ordered set partitions of [n] into k blocks such that equal-sized blocks are ordered with increasing least elements; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 6, 1, 0, 1, 11, 18, 1, 0, 1, 30, 75, 40, 1, 0, 1, 52, 420, 350, 75, 1, 0, 1, 126, 1218, 3080, 1225, 126, 1, 0, 1, 219, 4242, 17129, 15750, 3486, 196, 1, 0, 1, 510, 14563, 82488, 152355, 63756, 8526, 288, 1, 0, 1, 896, 42930, 464650, 1049895, 954387, 217560, 18600, 405, 1

Offset: 0

Alois P. Heinz, Apr 27 2017

			T(3,1) = 1: 123.
T(3,2) = 6: 1|23, 23|1, 2|13, 13|2, 3|12, 12|3.
T(3,3) = 1: 1|2|3.
Triangle T(n,k) begins:
  1;
  0, 1;
  0, 1,   1;
  0, 1,   6,    1;
  0, 1,  11,   18,     1;
  0, 1,  30,   75,    40,     1;
  0, 1,  52,  420,   350,    75,    1;
  0, 1, 126, 1218,  3080,  1225,  126,   1;
  0, 1, 219, 4242, 17129, 15750, 3486, 196, 1;
  ...

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

Columns k=0-10 give: A000007, A057427, A285917, A285918, A285919, A285920, A285921, A285922, A285923, A285924, A285925.
Main diagonal and first lower diagonal give: A000012, A002411.
Row sums give A120774.
T(2n,n) gives A285926.
Cf. A048993, A131689, A226874, A285849.

b:= proc(n, i, p) option remember; expand(`if`(n=0 or i=1,
      (p+n)!/n!*x^n, add(b(n-i*j, i-1, p+j)*x^j*combinat
      [multinomial](n, n-i*j, i$j)/j!^2, j=0..n/i)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n$2, 0)):
seq(T(n), n=0..12);

multinomial[n_, k_List] := n!/Times @@ (k!);
b[n_, i_, p_] := b[n, i, p] = Expand[If[n == 0 || i == 1, (p + n)!/n!*x^n, Sum[b[n-i*j, i-1, p+j]*x^j*multinomial[n, Join[{n-i*j}, Table[i, j]]]/ j!^2, {j, 0, n/i}]]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n, n, 0]];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Apr 28 2018, after Alois P. Heinz *)

cp's OEIS Frontend

A334997 Array T read by ascending antidiagonals: T(n, k) = Sum_{d divides n} T(d, k-1) with T(n, 0) = 1.

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A001499 Number of n X n matrices with exactly 2 1's in each row and column, other entries 0.

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A077592 Table by antidiagonals of tau_k(n), the k-th Piltz function (see A007425), or n-th term of the sequence resulting from applying the inverse Möbius transform (k-1) times to the all-ones sequence.

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A094727 Triangle read by rows: T(n,k) = n + k, 0 <= k < n, n >= 1.

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A093560 (3,1) Pascal triangle.

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A139601 Square array of polygonal numbers read by ascending antidiagonals: T(n, k) = (n + 1)*(k - 1)*k/2 + k.

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A139601 Square array of polygonal numbers read by ascending antidiagonals: T(n, k) = (n + 1)(k - 1)k/2 + k.

A033994 a(n) = n(n+1)(5*n+1)/6.