A005581 -id:A005581 - cp's OEIS frontend

A118404 Triangle T, read by rows, where all columns of T are different and yet all columns of the matrix square T^2 (A118407) are equal; also equals the matrix inverse of triangle A118400.

1, 1, -1, -1, 0, 1, -1, 1, -1, -1, 1, 0, 0, 2, 1, 1, -1, 0, -2, -3, -1, -1, 0, 1, 2, 5, 4, 1, -1, 1, -1, -3, -7, -9, -5, -1, 1, 0, 0, 4, 10, 16, 14, 6, 1, 1, -1, 0, -4, -14, -26, -30, -20, -7, -1, -1, 0, 1, 4, 18, 40, 56, 50, 27, 8, 1, -1, 1, -1, -5, -22, -58, -96, -106, -77, -35, -9, -1, 1, 0, 0, 6, 27, 80, 154, 202, 183, 112, 44, 10, 1, 1, -1, 0, -6, -33, -107, -234, -356, -385, -295, -156, -54, -11, -1, -1, 0, 1, 6, 39, 140, 341, 590, 741, 680, 451, 210, 65, 12, 1, -1, 1, -1, -7, -45, -179, -481, -931, -1331, -1421, -1131, -661, -275, -77, -13, -1, 1, 0, 0, 8, 52, 224, 660, 1412, 2262, 2752, 2552, 1792, 936, 352, 90, 14, 1

Offset: 0

Paul D. Hanna, Apr 27 2006

Appears to coincide with triangle (5.2) in Lee-Oh (2016), although there is no obvious connection! - N. J. A. Sloane, Dec 07 2016

			Triangle begins:
1;
1,-1;
-1, 0, 1;
-1, 1,-1,-1;
1, 0, 0, 2, 1;
1,-1, 0,-2,-3,-1;
-1, 0, 1, 2, 5, 4, 1;
-1, 1,-1,-3,-7,-9,-5,-1;
1, 0, 0, 4, 10, 16, 14, 6, 1;
1,-1, 0,-4,-14,-26,-30,-20,-7,-1;
-1, 0, 1, 4, 18, 40, 56, 50, 27, 8, 1;
-1, 1,-1,-5,-22,-58,-96,-106,-77,-35,-9,-1;
1, 0, 0, 6, 27, 80, 154, 202, 183, 112, 44, 10, 1;
1, -1, 0, -6, -33, -107, -234, -356, -385, -295, -156, -54, -11, -1;
-1, 0, 1, 6, 39, 140, 341, 590, 741, 680, 451, 210, 65, 12, 1;
-1, 1, -1, -7, -45, -179, -481, -931, -1331, -1421, -1131, -661, -275, -77, -13, -1;
1, 0, 0, 8, 52, 224, 660, 1412, 2262, 2752, 2552, 1792, 936, 352, 90, 14, 1;
1, -1, 0, -8, -60, -276, -884, -2072, -3674, -5014, -5304, -4344, -2728, -1288, -442, -104, -15, -1;
-1, 0, 1, 8, 68, 336, 1160, 2956, 5746, 8688, 10318, 9648, 7072, 4016, 1730, 546, 119, 16, 1; ...
The matrix square is A118407:
1;
0, 1;
-2, 0, 1;
2,-2, 0, 1;
0, 2,-2, 0, 1;
-2, 0, 2,-2, 0, 1;
4,-2, 0, 2,-2, 0, 1;
-6, 4,-2, 0, 2,-2, 0, 1;
4,-6, 4,-2, 0, 2,-2, 0, 1;
6, 4,-6, 4,-2, 0, 2,-2, 0, 1; ...
in which all columns are equal.

Kyu-Hwan Lee, Se-jin Oh, Catalan triangle numbers and binomial coefficients, arXiv:1601.06685 [math.CO], 2016.

Cf. A118405 (row sums), A118406 (unsigned row sums), A118407 (matrix square), A118400 (matrix inverse).

Columns or diagonals (modulo offsets): A219977, A011848, A212342, A007598, A005581, A007910.

T[n_, k_] := SeriesCoefficient[(-1)^k/((1+x^2)(1+x)^(k-1)), {x, 0, n-k}];
Table[T[n, k], {n, 0, 16}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 26 2018 *)

{T(n,k)=polcoeff(polcoeff((1+x)^2/(1+x^2)/(1+x+x*y +x*O(x^n)),n,x)+y*O(y^k),k,y)}
for(n=0, 16, for(k=0, n, print1(T(n, k), ", ")); print(""))

G.f.: A(x,y) = (1+x)^2 / ( (1+x^2) * (1+x + x*y) ).

G.f. of column k: (-1)^k / ( (1+x^2) * (1+x)^(k-1) ) for k>=0.

A121306 Array read by antidiagonals: a(m,n) = a(m,n-1)+a(m-1,n) but with initialization values a(0,0)=0, a(m>=1,0)=1, a(0,1)=1, a(0,n>1)=0.

Original entry on oeis.org

2, 2, 3, 2, 5, 4, 2, 7, 9, 5, 2, 9, 16, 14, 6, 2, 11, 25, 30, 20, 7, 2, 13, 36, 55, 50, 27, 8, 2, 15, 49, 91, 105, 77, 35, 9, 2, 17, 64, 140, 196, 182, 112, 44, 10, 19, 81, 204, 336, 378, 294, 156, 54, 100, 285, 540, 714, 672, 450, 210, 385, 825, 1254, 1386, 1122

Offset: 0

Thomas Wieder, Aug 04 2006, Aug 06 2006

For a(1,0)=1, a(m>1,0)=0 and a(0,n>=0)=0 one gets Pascal's triangle A007318.

			Array begins
2 2 2 2 2 2 2 2 2 ...
3 5 7 9 11 13 15 17 19 ...
4 9 16 25 36 49 64 81 100 ...
5 14 30 55 91 140 204 285 385 ...
6 20 50 105 196 336 540 825 1210 ...
7 27 77 182 378 714 1254 2079 3289 ...

The first nine rows are: A006527, A005408, A000290, A000330, A002415, A005585, A040977, A050486, A053347.
The initial columns are: A000027, A000096, A005581, A005582, A005583, A005584.
Cf. A119800, A007318, A006527, A005408, A000290, A000330, A002415, A005585, A040977, A050486, A053347, A000027, A000096, A005581, A005582, A005583, A005584.

=Z(-1)S+ZS(-1). The very first row (not included into the table) contains the initialization values: a(0,1)=1, a(0,n>=2)=0. The very first column (not included into the table) contains the initialization values: a(m>=1,0)=1. The value a(0,0)=0 does not enter into the table.

a(m,n) = a(m,n-1)+a(m-1,n), a(0,0)=0, a(m>=1,0)=1, a(0,1)=1, a(0,n>1)=0.

Edited by N. J. A. Sloane, Sep 15 2006

A185913 Accumulation array of A185912, by antidiagonals.

Original entry on oeis.org

1, 4, 6, 10, 21, 20, 20, 48, 66, 50, 35, 90, 144, 160, 105, 56, 150, 260, 340, 330, 196, 84, 231, 420, 600, 690, 609, 336, 120, 336, 630, 950, 1200, 1260, 1036, 540, 165, 468, 896, 1400, 1875, 2170, 2128, 1656, 825, 220, 630, 1224, 1960, 2730, 3360, 3640, 3384, 2520, 1210, 286, 825, 1620, 2640, 3780, 4851, 5600, 5760, 5130, 3685, 1716, 364, 1056, 2090, 3450, 5040, 6664, 8036, 8820, 8700, 7480, 5214, 2366, 455

Offset: 1

Clark Kimberling, Feb 06 2011

A member of the accumulation chain ... < A185910 < A185911 < A185912 < A185913 < ...

(See A144112 for definitions of weight array and accumulation array.)

			Northwest corner:
1.....4.....10.....20.....35
6.....21....48.....90.....150
20....66....144....260....420
50....160...340....600....950

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A144112, A185912.
Row 1 to 2: A000292, 3*A005581.
Column 1: A002415.

(* The program generates A185912 and its accumulation array A185913 *)
f[n_,k_]:=(k*n/6)(-2+3k+3n+2n^2);
TableForm[Table[f[n,k],{n,1,10},{k,1,15}]]  (* array A185912 *)
Table[f[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten
s[n_,k_]:=Sum[f[i,j],{i,1,n},{j,1,k}];
FullSimplify[s[n,k]]  (* formula for A185913 *)
TableForm[Table[s[n,k],{n,1,10},{k,1,15}]] (* array A185913 *)
Table[s[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten

T(n,k) = C(k+1,2)*C(n+1,2)*(n^2+3*n+2*k)/6, k>=1, n>=1.

A244422 Quasi-Riordan triangle ((2-z)/(1-z), -z^2/(1-z)). Row reversed monic Chebyshev T-polynomials without vanishing columns.

Original entry on oeis.org

2, 1, 0, 1, -2, 0, 1, -3, 0, 0, 1, -4, 2, 0, 0, 1, -5, 5, 0, 0, 0, 1, -6, 9, -2, 0, 0, 0, 1, -7, 14, -7, 0, 0, 0, 0, 1, -8, 20, -16, 2, 0, 0, 0, 0, 1, -9, 27, -30, 9, 0, 0, 0, 0, 0, 1, -10, 35, -50, 25, -2, 0, 0, 0, 0, 0, 1, -11, 44, -77, 55, -11, 0, 0, 0, 0, 0, 0, 1, -12, 54, -112, 105, -36, 2, 0, 0, 0, 0, 0, 0

Offset: 0

Wolfdieter Lang, Aug 08 2014

This is a signed version of the triangle A061896.
The coefficient table for the monic Chebyshev polynomials of the first kind R(n, x) = 2*T(n, x/2) is given in A127672. For the T-polynomials see A053120. The present table is obtained from the row reversed coefficient table A127672 by deleting all odd numbered columns which have only zeros, and appending in the rows numbered n >= 1 zeros in order to obtain a triangle. This becomes the quasi-Riordan triangle T = ((2-z)/(1-z), -z^2/(1-z)). This means that the o.g.f. of the row polynomials Rrev(n, x) := sqrt(x)^n*R(n, 1/sqrt(x)) = Sum_{k=0..n} T(n, k)*x^k have o.g.f. (2-z)/(1 - z + x*z^2) like for ordinary Riordan triangles. However this is not a Riordan triangle (or lower triangular infinite dimensional matrix) in the usual sense because it is not invertible. Therefore, this lower triangular matrix is not a member of the Riordan group.
The row sums give repeat(2,1,-1,-2,-1) which is A057079(n+1), n >= 0. The alternating row sums give the Lucas numbers A000032.

			The triangle T(n,k) begins:
  n\k  0   1   2     3    4     5  6   7  8  9 10 11
  0:   2
  1:   1   0
  2:   1  -2   0
  3:   1  -3   0     0
  4:   1  -4   2     0    0
  5:   1  -5   5     0    0     0
  6:   1  -6   9    -2    0     0  0
  7:   1  -7  14    -7    0     0  0   0
  8:   1  -8  20   -16    2     0  0   0  0
  9:   1  -9  27   -30    9     0  0   0  0  0
  10:  1 -10  35   -50   25    -2  0   0  0  0  0
  11:  1 -11  44   -77   55   -11  0   0  0  0  0  0
  ...
Rrev(3, x) = 1 - 3*x = sqrt(x)^3*R(3,1/sqrt(x)) = sqrt(x)^3*(-3/sqrt(x) + 1/sqrt(x)^3 ) = -3*x + 1.
Rrev(4, x) = 1 - 4*x + 2*x^2 = sqrt(x)^4*(2 - 4/sqrt(x)^2 + 1/sqrt(x)^4) = 2*x^2 - 4*x + 1.
Recurrence: T(4,1) = T(3, 1) - T(2, 0) = -3 -1 = -4.

Wolfdieter Lang, First rows of the triangle.

Cf. A053120, A127672, A054977, -A000027, A000096, -A005581, A005582, -A005583, A005584, A057079, A000032.

T(n,k) = [x^k] Rrev(n, x), k=0, 1, ..., n, with the row polynomials Rrev(n, x) = sqrt(x)^n*R(n,1/sqrt(x)), with R(n, x) given in A127672 (monic Chebyshev polynomials of the first kind).
O.g.f. row polynomials Rrev(n,x) = Sum_{k=0..n} T(n,k)*x^k: (2-z)/(1 - z + x*z^2) (quasi-Riordan).
O.g.f. for column number k entries with leading zeros: ((2-x)/(1-x))*(-x^2/(1-x))^k, k > = 0. See A054977, -A000027, A000096, -A005581, A005582, -A005583, A005584.
Recurrence: T(n,k) = T(n-1, k) - T(n-2, k-1), n >= k >= 1, T(n,k) = 0 if n < k, T(0,0) = 2, T(n,0) = 1 if n>=1, (Compare with A061896).
For n >= 1 the entries without trailing zeros are given by T(n,k) = (-1)^k*(n/(n-k))*binomial(n-k,k) where k=0..floor(n/2).

A208518 Triangle of coefficients of polynomials u(n,x) jointly generated with A208519; see the Formula section.

Original entry on oeis.org

1, 1, 1, 1, 3, 2, 1, 6, 7, 3, 1, 10, 16, 14, 5, 1, 15, 30, 40, 28, 8, 1, 21, 50, 90, 93, 53, 13, 1, 28, 77, 175, 238, 203, 99, 21, 1, 36, 112, 308, 518, 588, 428, 181, 34, 1, 45, 156, 504, 1008, 1428, 1380, 873, 327, 55, 1, 55, 210, 780, 1806, 3066, 3690, 3105

Offset: 1

Clark Kimberling, Feb 28 2012

coefficient of x^(n-1): = Fibonacci(n) = A000045(n)
col 1:  A000012
col 2:  A000217 (triangular numbers)
col 3:  A005581
col 4:  A117662
alternating row sums: signed version of (-1+Fibonacci(n))

			First five rows:
1
1...1
1...3....2
1...6....7....3
1...10...16...14...5
First five polynomials u(n,x):
1
1 + x
1 + 3x + 2x^2
1 + 6x + 7x^2 + 3x^3
1 + 10x + 16x^2 + 14x^3 + 5x^4

Cf. A208519.

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
v[n_, x_] := x*u[n - 1, x] + (x + 1)*v[n - 1, x] + 1;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]   (* A208518 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]   (* A208519 *)

u(n,x)=u(n-1,x)+x*v(n-1,x),
v(n,x)=x*u(n-1,x)+(x+1)*v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.

A254749 1-gonal pyramidal numbers.

Original entry on oeis.org

1, 2, 2, 0, -5, -14, -28, -48, -75, -110, -154, -208, -273, -350, -440, -544, -663, -798, -950, -1120, -1309, -1518, -1748, -2000, -2275, -2574, -2898, -3248, -3625, -4030, -4464, -4928, -5423, -5950, -6510, -7104, -7733, -8398, -9100, -9840, -10619, -11438

Offset: 1

Colin Barker, Feb 07 2015

Not strictly pyramidal numbers, but the result of using the Wikipedia formula with r = 1.

Essentially the same as A129936 and A005586.

			G.f. = x + 2*x^2 + 2*x^3 - 5*x^5 - 14*x^6 - 28*x^7 - 48*x^8 - 75*x^9 + ...

Colin Barker, Table of n, a(n) for n = 1..1000
Wikipedia, Pyramidal number.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A005581, A005586, A129936.

[(n*(4+3*n-n^2))/6: n in [1..60]]; // G. C. Greubel, Aug 03 2018

Table[(n*(4+3*n-n^2))/6, {n,1,60}] (* or *) LinearRecurrence[{4,-6,4,-1}, {1, 2, 2, 0}, 60] (* G. C. Greubel, Aug 03 2018 *)

ppg(r, n) = (3*n^2+n^3*(r-2)-n*(r-5))/6
vector(100, n, ppg(1, n))

a(n) = n*(4 + 3*n - n^2)/6.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
G.f.: x*(1 - 2*x)/(x-1)^4.
a(n) = A005581(-n) = -A005586(n-4) = -A129936(n-2) for all n in Z. - Michael Somos, Jul 28 2015
E.g.f.: -exp(x)*x*(x^2 - 6)/6. - Elmo R. Oliveira, Aug 04 2025

A264357 Array A(r, n) of number of independent components of a symmetric traceless tensor of rank r and dimension n, written as triangle T(n, r) = A(r, n-r+2), n >= 1, r = 2..n+1.

Original entry on oeis.org

0, 2, 0, 5, 2, 0, 9, 7, 2, 0, 14, 16, 9, 2, 0, 20, 30, 25, 11, 2, 0, 27, 50, 55, 36, 13, 2, 0, 35, 77, 105, 91, 49, 15, 2, 0, 44, 112, 182, 196, 140, 64, 17, 2, 0, 54, 156, 294, 378, 336, 204, 81, 19, 2, 0

Offset: 1

Wolfdieter Lang, Dec 10 2015

A (totally) symmetric traceless tensor of rank r >= 2 and dimension n >= 1 is irreducible.
The array of the number of independent components of a rank r symmetric traceless tensor A(r, n), for r >= 2 and n >=1, is given by risefac(n,r)/r! - risefac(n,r-2)/(r-2)!, where the first term gives the number of independent components of a symmetric tensors of rank r (see a Dec 10 2015 comment under A135278) and the second term is the number of constraints from the tracelessness requirement. The tensor has to be traceless in each pair of indices.
The first rows of the array A, or the first columns (without the first r-2 zeros) of the triangle T are for r = 2..6: A000096, A005581, A005582, A005583, A005584.
Equals A115241 with the first column of positive integers removed. - Georg Fischer, Jul 26 2023

			The array A(r, n) starts:
   r\n 1 2  3   4   5    6    7     8     9    10 ...
   2:  0 2  5   9  14   20   27    35    44    54
   3:  0 2  7  16  30   50   77   112   156   210
   4:  0 2  9  25  55  105  182   294   450   660
   5:  0 2 11  36  91  196  378   672  1122  1782
   6:  0 2 13  49 140  336  714  1386  2508  4290
   7:  0 2 15  64 204  540 1254  2640  5148  9438
   8:  0 2 17  81 285  825 2079  4719  9867 19305
   9:  0 2 19 100 385 1210 3289  8008 17875 37180
  10:  0 2 21 121 506 1716 5005 13013 30888 68068
  ...
The triangle T(n, r) starts:
   n\r  2   3   4   5   6   7  8  9 10 11 ...
   1:   0
   2:   2   0
   3:   5   2   0
   4:   9   7   2   0
   5:  14  16   9   2   0
   6:  20  30  25  11   2   0
   7:  27  50  55  36  13   2  0
   8:  35  77 105  91  49  15  2  0
   9:  44 112 182 196 140  64 17  2  0
  10:  54 156 294 378 336 204 81 19  2  0
  ...
A(r, 1) = 0 , r >= 2, because a symmetric rank r tensor t of dimension one has one component t(1,1,...,1) (r 1's) and if the traces vanish then t vanishes.
A(3, 2) = 2 because a symmetric rank 3 tensor t with three indices taking values from 1 or 2 (n=2) has the four independent components t(1,1,1), t(1,1,2), t(1,2,2), t(2,2,2), and (invoking symmetry) the vanishing traces are Sum_{j=1..2} t(j,j,1) = 0 and Sum_{j=1..2} t(j,j,2) = 0. These are two constraints, which can be used to eliminate, say, t(1,1,1) and t(2,2,2), leaving 2 = A(3, 2) independent components, say, t(1,1,2) and t(1,2,2).
From _Peter Luschny_, Dec 14 2015: (Start)
The diagonals diag(n, k) start:
   k\n  0       1       2       3       4      5       6
   0:   0,      2,      9,     36,    140,   540,   2079, ... A007946
   1:   2,      7,     25,     91,    336,  1254,   4719, ... A097613
   2:   5,     16,     55,    196,    714,  2640,   9867, ... A051960
   3:   9,     30,    105,    378,   1386,  5148,  19305, ... A029651
   4:  14,     50,    182,    672,   2508,  9438,  35750, ... A051924
   5:  20,     77,    294,   1122,   4290, 16445,  63206, ... A129869
   6:  27,    112,    450,   1782,   7007, 27456, 107406, ... A220101
   7:  35,    156,    660,   2717,  11011, 44200, 176358, ... A265612
   8:  44,    210,    935,    4004, 16744, 68952, 281010, ... A265613
  A000096,A005581,A005582,A005583,A005584.
(End)

Cf. A000096, A005581, A005582, A005583, A005584, A007946, A024482, A029651, A051924, A051960, A097613, A115241, A129869, A135278, A220101, A265612, A265613.

A[r_, n_] := Pochhammer[n, r]/r! - Pochhammer[n, r-2]/(r-2)!;
T[n_, r_] := A[r, n-r+2];
Table[T[n, r], {n, 1, 10}, {r, 2, n+1}] (* Jean-François Alcover, Jun 28 2019 *)

A = lambda r, n: rising_factorial(n,r)/factorial(r) - rising_factorial(n,r-2)/factorial(r-2)
for r in (2..10): [A(r,n) for n in (1..10)] # Peter Luschny, Dec 13 2015

T(n, r) = A(r, n-r+2) with the array A(r, n) = risefac(n,r)/r! - risefac(n,r-2)/(r-2)! where the rising factorial risefac(n,k) = Product_{j=0..k-1} (n+j) and risefac(n,0) = 1.
From Peter Luschny, Dec 14 2015: (Start)
A(n+2, n+1) = A007946(n-1) = CatalanNumber(n)*3*n*(n+1)/(n+2) for n>=0.
A(n+2, n+2) = A024482(n+2) = A097613(n+2) = CatalanNumber(n+1)*(3*n+4)/2 for n>=0.
A(n+2, n+3) = A051960(n+1) = CatalanNumber(n+1)*(3*n+5) for n>=0.
A(n+2, n+4) = A029651(n+2) = CatalanNumber(n+1)*(6*n+9) for n>=0.
A(n+2, n+5) = A051924(n+3) = CatalanNumber(n+2)*(3*n+7) for n>=0.
A(n+2, n+6) = A129869(n+4) = CatalanNumber(n+2)*(3*n+8)*(2*n+5)/(n+4) for n>=0.
A(n+2, n+7) = A220101(n+4) = CatalanNumber(n+3)*(3*(n+3)^2)/(n+5) for n>=0.
A(n+2, n+8) = CatalanNumber(n+4)*(n+3)*(3*n+10)/(2*n+12) for n>=0.
Let for n>=0 and k>=0 diag(n,k) = A(k+2,n+k+1) and G(n,k) = 2^(k+2*n)*Gamma((3-(-1)^k+2*k+4*n)/4)/(sqrt(Pi)*Gamma(k+n+0^k)) then
diag(n,0) = G(n,0)*(n*3)/(n+2),
diag(n,1) = G(n,1)*(3*n+4)/((n+1)*(n+2)),
diag(n,2) = G(n,2)*(3*n+5)/(n+2),
diag(n,3) = G(n,3)*3,
diag(n,4) = G(n,4)*(3*n+7),
diag(n,5) = G(n,5)*(3*n+8),
diag(n,6) = G(n,6)*3*(3+n)^2,
diag(n,7) = G(n,7)*(3+n)*(10+3*n). (End)

A185783 Second accumulation array of A185780, by antidiagonals.

Original entry on oeis.org

1, 6, 3, 20, 20, 6, 50, 70, 44, 10, 105, 180, 160, 80, 15, 196, 385, 420, 300, 130, 21, 336, 728, 910, 800, 500, 196, 28, 540, 1260, 1736, 1750, 1350, 770, 280, 36, 825, 2040, 3024, 3360, 2975, 2100, 1120, 384, 45, 1210, 3135, 4920, 5880, 5740, 4655, 3080, 1560, 510, 55, 1716, 4620, 7590, 9600, 10080, 9016, 6860, 4320, 2100, 660, 66, 2366, 6578, 11220, 14850, 16500, 15876, 13328, 9660, 5850, 2750, 836, 78

Offset: 1

Clark Kimberling, Feb 03 2011

See A144112 and A185780.

			Northwest corner:
1....6....20....50....105
3....20...70....180...385
6....44...160...420...910
10...80...300...800...1750

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A144112, A185780.
Row 1: A002415 (4-dimensional pyramidal numbers).
Columns 1 to 3: A000217, 2*A006503, 10*A005581.

(See A185780.)
f[n_, k_] := Binomial[k + 2, 3]*Binomial[n + 1, 2]*(k*n - n + 2*k + 4)/6; Table[f[n - k + 1, k], {n, 10}, {k, n, 1, -1}] // Flatten (* G. C. Greubel, Jul 12 2017 *)

T(n,k) = C(k+2,3)*C(n+1,2)*(k*n-n+2*k+4)/6, k>=1, n>=1.

A349417 a(n) is the Wiener index of a sling on n+1 vertices.

Original entry on oeis.org

9, 18, 32, 52, 79, 114, 158, 212, 277, 354, 444, 548, 667, 802, 954, 1124, 1313, 1522, 1752, 2004, 2279, 2578, 2902, 3252, 3629, 4034, 4468, 4932, 5427, 5954, 6514, 7108, 7737, 8402, 9104, 9844, 10623, 11442, 12302, 13204, 14149, 15138, 16172, 17252, 18379, 19554, 20778

Offset: 3

Julian Allagan, Nov 16 2021

A sling on n+1 vertices is a caterpillar that is obtained by adding 1 pendant vertex to the first (or last) internal vertex of a path on n >= 3 vertices.

			For n=3, a(3)=9 gives the Wiener index of a star graph on 4 vertices. For n=4, a(4)=18 gives the Wiener index of a sling graph on 5 vertices.
   *
*__\*__*__*

Cf. A349416 (broom), A349418 (tridon).

Essentially same as A005581(n)+2.

Table[n^3/6 + n^2/2 - 2n/3 + 2, {n, 3, 102}]

a(n) = n^3/6 + n^2/2 - 2n/3 + 2.

A383333 Square array read by antidiagonals: T(n,k) is the number of n-tuples of nonnegative integers, not all equal to 0, with a shortest vectorial addition chain of length k; n >= 1, k >= 0.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 3, 7, 6, 4, 5, 16, 16, 10, 5, 9, 37, 46, 30, 15, 6, 15, 91, 134, 101, 50, 21, 7, 26, 229, 411, 349, 190, 77, 28, 8, 44, 585, 1319, 1264, 751, 323, 112, 36, 9, 78, 1528, 4368, 4817, 3106, 1426, 511, 156, 45, 10, 136, 4034, 14925, 19131, 13532, 6586, 2478, 766, 210, 55, 11

Offset: 1

Pontus von Brömssen, Apr 26 2025

The n unit tuples (1, 0, ..., 0), ... (0, ..., 0, 1) are given for free, so T(n,0) = n.
Starting with the n unit tuples, each tuple of a chain must be equal to the sum of two preceding tuples. The length of the chain is defined to be the number of tuples in the chain, excluding the unit tuples.
Also, T(n,k) is the number of non-constant monomials x_1^e_1*...*x_n^e_n that requires k multiplications, given x_1, ..., x_n.

			Array begins:
  n\k| 0  1  2   3    4    5     6      7
  ---+-----------------------------------
  1  | 1  1  2   3    5    9    15     26
  2  | 2  3  7  16   37   91   229    585
  3  | 3  6 16  46  134  411  1319   4368
  4  | 4 10 30 101  349 1264  4817  19131
  5  | 5 15 50 190  751 3106 13532  61748
  6  | 6 21 77 323 1426 6586 32035 163594
There are 12 triples of nondecreasing nonnegative integers with a shortest addition chain of length 3. Counting also the permutations of these, we get T(3,3) = 46:
  (0, 0, 5):  3
  (0, 0, 6):  3
  (0, 0, 8):  3
  (0, 1, 3):  6
  (0, 1, 4):  6
  (0, 2, 3):  6
  (0, 2, 4):  6
  (0, 3, 3):  3
  (0, 4, 4):  3
  (1, 1, 2):  3
  (1, 2, 2):  3
  (2, 2, 2):  1
      Total: 46

Jorge Olivos, On vectorial addition chains, Journal of Algorithms 2 (1981), 13-21.
E. G. Straus, Partial solution to problem 5125: Addition chains of vectors, Problems and solutions, The American Mathematical Monthly 71 (1964), 806-808.
Edward G. Thurber and Neill M. Clift, Addition chains, vector chains, and efficient computation, Discrete Mathematics 344 (2021), 112200.
Wikipedia, Vectorial addition chain.

Cf. A383334.
Rows: A003065 (n=1), A383331 (n=2).
Columns: A000027 (k=0), A000217 (k=1), A005581 (k=2).

cp's OEIS Frontend

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