A000096 -id:A000096 - cp's OEIS frontend

A061579 Reverse one number (0), then two numbers (2,1), then three (5,4,3), then four (9,8,7,6), etc.

0, 2, 1, 5, 4, 3, 9, 8, 7, 6, 14, 13, 12, 11, 10, 20, 19, 18, 17, 16, 15, 27, 26, 25, 24, 23, 22, 21, 35, 34, 33, 32, 31, 30, 29, 28, 44, 43, 42, 41, 40, 39, 38, 37, 36, 54, 53, 52, 51, 50, 49, 48, 47, 46, 45, 65, 64, 63, 62, 61, 60, 59, 58, 57, 56, 55, 77, 76, 75, 74, 73, 72, 71, 70, 69, 68, 67, 66

Offset: 0

Henry Bottomley, May 21 2001

A self-inverse permutation of the nonnegative numbers.
a(n) is the smallest nonnegative integer not yet in the sequence such that n + a(n) is one less than a square. - Franklin T. Adams-Watters, Apr 06 2009
From Michel Marcus, Mar 01 2021: (Start)
Array T(n,k) = (n+k)^2/2 + (n+3*k)/2 for n,k >= 0 read by descending antidiagonals.
Array T(n,k) = (n+k)^2/2 + (3*n+k)/2 for n,k >= 0 read by ascending antidiagonals. (End)

			Read as a triangle, the sequence is:
    0
    2   1
    5   4   3
    9   8   7   6
   14  13  12  11  10
  (...)
As an infinite square matrix (cf. the "table" link, 2nd paragraph) it reads:
    0    2    5    9   14   20   ...
    1    4    8   13   19   22   ...
    3    7   12   18   23   30   ...
    6   11   17   24   31   39   ...
  (...)

Harry J. Smith, Table of n, a(n) for n = 0..1000
Madeline Brandt and Kåre Schou Gjaldbæk, Classification of Quadratic Packing Polynomials on Sectors of R^2, arXiv:2102.13578 [math.NT], 2021. See Figure 9 p. 17.
Gennady Eremin, Partitioning the set of natural numbers into Mersenne trees and into arithmetic progressions; Natural Matrix and Linnik's constant, arXiv:2405.16143 [math.CO], 2024.
Index entries for sequences that are permutations of the natural numbers

Fixed points are A046092.
Row sums give A027480.
Each reversal involves the numbers from A000217 through to A000096.
Cf. A038722. Transpose of A001477.
Cf. A002419, A119771, A226725.

T:= (n,k)-> n*(n+3)/2-k:
seq(seq(T(n,k), k=0..n), n=0..12);  # Alois P. Heinz, Feb 10 2023

Module[{nn=20},Reverse/@TakeList[Range[0,(nn(nn+1))/2],Range[nn]]]// Flatten (* Requires Mathematica version 11 or later *) (* Harvey P. Dale, Jul 06 2018 *)

A061579_row(n)=vector(n+=1, j, n*(n+1)\2-j)
A061579_upto(n)=concat([A061579_row(r)|r<-[0..sqrtint(2*n)]]) \\ yields approximately n terms: actual number differs by less than +- sqrt(n). - M. F. Hasler, Nov 09 2021

from math import isqrt
def A061579(n): return (r:=isqrt((n<<3)+1)-1>>1)*(r+2)-n # Chai Wah Wu, Feb 10 2023

a(n) = floor(sqrt(2n+1)-1/2)*floor(sqrt(2n+1)+3/2) - n = A005563(A003056(n)) - n.
Row (or antidiagonal) n = 0, 1, 2, ... contains the integers from A000217(n) to A000217(n+1)-1 in reverse order (for diagonals, "reversed" with respect to the canonical "falling" order, cf. A001477/table). - M. F. Hasler, Nov 09 2021
From Alois P. Heinz, Feb 10 2023: (Start)
T(n,k) = n*(n+3)/2 - k.
Sum_{k=0..n} k * T(n,k) = A002419(n).
Sum_{k=0..n} k^2 * T(n,k) = A119771(n).
Sum_{k=0..n} (-1)^k * T(n,k) = A226725(n). (End)

A126890 Triangle read by rows: T(n,k) = n(n+2k+1)/2, 0 <= k <= n.

Original entry on oeis.org

0, 1, 2, 3, 5, 7, 6, 9, 12, 15, 10, 14, 18, 22, 26, 15, 20, 25, 30, 35, 40, 21, 27, 33, 39, 45, 51, 57, 28, 35, 42, 49, 56, 63, 70, 77, 36, 44, 52, 60, 68, 76, 84, 92, 100, 45, 54, 63, 72, 81, 90, 99, 108, 117, 126, 55, 65, 75, 85, 95, 105, 115, 125, 135, 145, 155, 66, 77, 88

Offset: 0

Reinhard Zumkeller, Dec 30 2006

T(n,k) + T(n,n-k) = A014105(n);
row sums give A059270; Sum_{k=0..n-1} T(n,k) = A000578(n);
central terms give A007742; T(2*n+1,n) = A016754(n);
T(n,0)   = A000217(n);
T(n,1)   = A000096(n)   for n > 0;
T(n,2)   = A055998(n)   for n > 1;
T(n,3)   = A055999(n)   for n > 2;
T(n,4)   = A056000(n)   for n > 3;
T(n,5)   = A056115(n)   for n > 4;
T(n,6)   = A056119(n)   for n > 5;
T(n,7)   = A056121(n)   for n > 6;
T(n,8)   = A056126(n)   for n > 7;
T(n,10)  = A101859(n-1) for n > 9;
T(n,n-3) = A095794(n-1) for n > 2;
T(n,n-2) = A045943(n-1) for n > 1;
T(n,n-1) = A000326(n)   for n > 0;
T(n,n)   = A005449(n).

			From _Philippe Deléham_, Oct 03 2011: (Start)
Triangle begins:
   0;
   1,  2;
   3,  5,  7;
   6,  9, 12, 15;
  10, 14, 18, 22, 26;
  15, 20, 25, 30, 35, 40;
  21, 27, 33, 39, 45, 51, 57;
  28, 35, 42, 49, 56, 63, 70, 77; (End)

Léonard Euler, Introduction à l'analyse infinitésimale, tome premier, ACL-Editions, Paris, 1987, p. 353-354.

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
Émile Fourrey, Les nombres abstraits, Récreations arithmétiques, 1899 and later, Vuibert, Paris, page 86-87. Triangle without right diagonal.
Adrien-Marie Legendre, Théorie des nombres, tome 2, quatrième partie, p.131, troisième édition, Paris, 1830.

Cf. A110449.

a126890 n k = a126890_tabl !! n !! k
a126890_row n = a126890_tabl !! n
a126890_tabl = map fst $ iterate
   (\(xs@(x:_), i) -> (zipWith (+) ((x-i):xs) [2*i+1 ..], i+1)) ([0], 0)
-- Reinhard Zumkeller, Nov 10 2013

Flatten[Table[(n(n+2k+1))/2,{n,0,20},{k,0,n}]] (* Harvey P. Dale, Jun 21 2013 *)

T(n,k) = T(n,k-1) + n, for k <= n. - Philippe Deléham, Oct 03 2011

A185787 Sum of first k numbers in column k of the natural number array A000027; by antidiagonals.

Original entry on oeis.org

1, 7, 25, 62, 125, 221, 357, 540, 777, 1075, 1441, 1882, 2405, 3017, 3725, 4536, 5457, 6495, 7657, 8950, 10381, 11957, 13685, 15572, 17625, 19851, 22257, 24850, 27637, 30625, 33821, 37232, 40865, 44727, 48825, 53166, 57757, 62605, 67717, 73100, 78761, 84707, 90945, 97482, 104325, 111481, 118957, 126760, 134897, 143375

Offset: 1

Clark Kimberling, Feb 03 2011

This is one of many interesting sequences and arrays that stem from the natural number array A000027, of which a northwest corner is as follows:
  1....2.....4.....7...11...16...22...29...
  3....5.....8....12...17...23...30...38...
  6....9....13....18...24...31...39...48...
  10...14...19....25...32...40...49...59...
  15...20...26....33...41...50...60...71...
  21...27...34....42...51...61...72...84...
  28...35...43....52...62...73...85...98...
Blocking out all terms below the main diagonal leaves columns whose sums comprise A185787.  Deleting the main diagonal and then summing give A185787.  Analogous treatments to the left of the main diagonal give A100182 and A101165.  Further sequences obtained directly from this array are easily obtained using the following formula for the array: T(n,k)=n+(n+k-2)(n+k-1)/2.
Examples:
row 1:  A000124
row 2:  A022856
row 3:  A016028
row 4:  A145018
row 5:  A077169
col 1:  A000217
col 2:  A000096
col 3:  A034856
col 4:  A055998
col 5:  A046691
col 6:  A052905
col 7:  A055999
diag. (1,5,...) ...... A001844
diag. (2,8,...) ...... A001105
diag. (4,12,...)...... A046092
diag. (7,17,...)...... A056220
diag. (11,23,...) .... A132209
diag. (16,30,...) .... A054000
diag. (22,38,...) .... A090288
diag. (3,9,...) ...... A058331
diag. (6,14,...) ..... A051890
diag. (10,20,...) .... A005893
diag. (15,27,...) .... A097080
diag. (21,35,...) .... A093328
antidiagonal sums:  (1,5,15,34,...)=A006003=partial sums of A002817.
Let S(n,k) denote the n-th partial sum of column k.  Then
S(n,k)=n*(n^2+3k*n+3*k^2-6*k+5)/6.
S(n,1)=n(n+1)(n+2)/6
S(n,2)=n(n+1)(n+5)/6
S(n,3)=n(n+2)(n+7)/6
S(n,4)=n(n^2+12n+29)/6
S(n,5)=n(n+5)(n+10)/6
S(n,6)=n(n+7)(n+11)/6
S(n,7)=n(n+10)(n+11)/6
Weight array of T: A144112
Accumulation array of T: A185506
Second rectangular sum array of T: A185507
Third rectangular sum array of T: A185508
Fourth rectangular sum array of T: A185509

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

Cf. A000027, A185788, A100182, A101165, A079824.

[n*(7*n^2-6*n+5)/6: n in [1..50]]; // Vincenzo Librandi, Jul 04 2012

f[n_,k_]:=n+(n+k-2)(n+k-1)/2;
s[k_]:=Sum[f[n,k],{n,1,k}];
Factor[s[k]]
Table[s[k],{k,1,70}]  (* A185787 *)
CoefficientList[Series[(3*x^2+3*x+1)/(1-x)^4,{x,0,50}],x] (* Vincenzo Librandi, Jul 04 2012 *)

a(n)=n*(7*n^2-6*n+5)/6.

G.f.: x*(3*x^2+3*x+1)/(1-x)^4. - Vincenzo Librandi, Jul 04 2012

Edited by Clark Kimberling, Feb 25 2023

A111774 Numbers that can be written as a sum of at least three consecutive positive integers.

Original entry on oeis.org

6, 9, 10, 12, 14, 15, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 102

Offset: 1

Jaap Spies, Aug 15 2005

In this sequence there are no (odd) primes and there are no powers of 2.
So we have only three kinds of natural numbers: the odd primes, the powers of 2 and the numbers that can be represented as a sum of at least three consecutive integers.
Odd primes can only be written as a sum of two consecutive integers. Powers of 2 do not have a representation as a sum of k consecutive integers (other than the trivial n=n, for k=1).
Numbers of the form (x*(x+1)-y*(y+1))/2 for nonnegative integers x,y with x-y >= 3. - Bob Selcoe, Feb 21 2014
Numbers of the form (x + 1)*(x + 2*y)/2 for integers x,y with x >= 2 and y >= 1. For y = 1 only triangular numbers (A000217) >= 6 occur. - Ralf Steiner, Jun 27 2019
From Ralf Steiner, Jul 09 2019: (Start)
If k >= 1 sequences are c_k(n) = c_k(n - 1) + n + k - 1, c_k(0) = 0, means c_k(n) = n*(n + 2*k - 1)/2: A000217, A000096, A055998, A055999, A056000, ... then this sequence is the union of c_k(n), n >= 3. (End)
From Wolfdieter Lang, Oct 28 2020: (Start)
This sequence gives all positive integers that have at least one odd prime as proper divisor. The proof follows from the first two comments.
The set {a(n)}_{n>=1} equals the set {k positive integer : floor(k/2) - delta(k) >= 1}, where delta(k) = A055034(k). Proof: floor(k/2) gives the number of positive odd numbers < k, and delta(k), gives the number of positive odd numbers coprime to k. delta(1) = 1 but 1 is not < 1, therefore k = 1 is not a member of this set. Hence a member >= 2 of this set has at least one odd number > 1 and < k missing in the set of odd numbers relative prime to k. Therefore there exists at least one odd prime < k dividing k. (End)
For the multiplicity of a(n) see A338428, obtained from triangle A337940 (the array is given Bob Selcoe as example below, and in the Ralf Steiner comment above). - Wolfdieter Lang, Dec 09 2020

			a(1)=6 because 6 is the first number that can be written as a sum of three consecutive positive integers: 6 = 1+2+3.
From _Bob Selcoe_, Feb 23 2014: (Start)
Let the top row of an array be A000217(n). Let the diagonals (reading down and left) be A000217(n)-A000217(1),  A000217(n)-A000217(2), A000217(n)-A000217(3)..., A000217(n)-A000217(n-3).  This is A049777 read as a square array, starting with the third column. The array begins as follows:
   6 10 15 21 28 36 45 55 66
   9 14 20 27 35 44 54 65
  12 18 25 33 42 52 63
  15 22 30 39 49 60
  18 26 35 45 56
  21 30 40 51
  24 34 45
  27 38
  30
This is (x*(x+1)-y*(y+1))/2 for nonnegative integers x,y with x-y >= 3, because it is equivalent to 1+2+3/+4/+5/...+x/-0/-1/-2/-3/-4/-5/...-(x+3)/ for all possible strings of consecutive integers, which represents every possible way to sum three or more consecutive positive integers. So for example, 4+5+6+7 = 1+2+3+4+5+6+7-1-2-3 = 22, which is (x*(x+1)-y*(y+1))/2 when x=7, y=3. Notice that values can appear more than once in the array because some numbers can be represented as sums of more than one string of three or more consecutive positive integers. For example, 30 = (x*(x+1)-y*(y+1))/2 when (a) x=11, y=8: 9+10+11; (b) x=9, y=5: 6+7+8+9; and (c) x=8, y=3: 4+5+6+7+8. By definition, x-y is the number of integers in the string. (End)

Paul Halmos, "Problems for Mathematicians, Young and Old", Dolciani Mathematical Expositions, 1991, Solution to problem 3G p. 179.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Nieuw Archief voor Wiskunde 5/6 nr. 2 Problems/UWC, Problem C, Jun 2005, p. 181-182.
Nieuw Archief voor Wiskunde 5/6 nr. 2 Problems/UWC, Problem C: Solution of this Problem
J. Spies, Sage program for computing A111774

Cf. A000040, A000079, A066542, A174090 (complement), A000217, A049777, A055034.

Cf. A337940, A338428.

ispoweroftwo := proc(n) local a, t; t := 1; while (n > t) do t := 2*t end do; if (n = t) then a := true else a := false end if; return a; end proc; f:= proc(n) if (not isprime(n)) and (not ispoweroftwo(n)) then return n end if; end proc; seq(f(i),i = 1..150);

max=6!;lst={};Do[z=n+(n+1);Do[z+=(n+x);If[z>max,Break[]];AppendTo[lst,z],{x,2,max}],{n,max}];Union[lst] (* Vladimir Joseph Stephan Orlovsky, Mar 06 2010 *)

isok(n) = !(n == 1) && !isprime(n) && !(isprimepower(n, &p) && (p == 2)); \\ Michel Marcus, Jul 02 2019

from sympy import primepi
def A111774(n):
    def f(x): return int(n+(0 if x<=1 else primepi(x)-1)+x.bit_length())
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Sep 19 2024

A046691 a(n) = (n^2 + 5*n - 2)/2.

Original entry on oeis.org

-1, 2, 6, 11, 17, 24, 32, 41, 51, 62, 74, 87, 101, 116, 132, 149, 167, 186, 206, 227, 249, 272, 296, 321, 347, 374, 402, 431, 461, 492, 524, 557, 591, 626, 662, 699, 737, 776, 816, 857, 899, 942, 986, 1031, 1077, 1124, 1172, 1221, 1271, 1322

Offset: 0

N. J. A. Sloane

If Y_i (i=1,2,3,4) are 2-blocks of an n-set X then, for n>=8, a(n-3) is the number of (n-2)-subsets of X intersecting each Y_i (i=1,2,3,4). - Milan Janjic, Nov 09 2007
Numbers m > -3 such that 8*m + 33 is a square. - Bruno Berselli, Aug 20 2015
a(n-1) yields the second Betti number of a path graph on n vertices. - Samuel J. Bevins, Nov 27 2022

G. C. Greubel, Table of n, a(n) for n = 0..5000
Milan Janjic, Two Enumerative Functions
P. Di Francesco, O. Golinelli and E. Guitter, Meander, folding and arch statistics, arXiv:hep-th/9506030, 1995.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Triangular numbers (A000217) minus 4. Cf. A027379.

Cf. A000096, A000217, A002522, A028884.

[Binomial(n+3,2) -4: n in [0..50]]; // G. C. Greubel, Jul 31 2022

seq(binomial(n+3, 2)-4, n=0..49); # Zerinvary Lajos, Jan 13 2007

Table[(n^2 +5n -2)/2, {n, 0, 50}] (* Bruno Berselli, Dec 17 2014 *)

a(n)=(n^2+5*n-2)/2 \\ Charles R Greathouse IV, Oct 07 2015

[(n^2 +5*n -2)/2 for n in (0..50)] # G. C. Greubel, Jul 31 2022

G.f.: (-1 + 5*x - 3*x^2)/(1 - x)^3.
a(n) = a(n-1) + n + 2 with a(0) = -1. - Vincenzo Librandi, Nov 18 2010
a(n) = 3*A000096(n-1) - 2*A000096(n-2), with A000096(-2)=A000096(-1)=-1. - Bruno Berselli, Dec 17 2014
a(n) = 2*A000217(n) - A000217(n-2), with A000217(-2)=1, A000217(-1)=0. - Bruno Berselli, Oct 13 2016
E.g.f.: (1/2)*(x^2 + 6*x - 2)*exp(x). - G. C. Greubel, Jul 13 2017
a(n) + a(n+1) = A028884(n). - R. J. Mathar, Mar 13 2021
a(n) = A000217(n+2) - 4. - G. C. Greubel, Jul 31 2022
Sum_{n>=0} 1/a(n) = 7/12 + 2*Pi*tan(sqrt(33)*Pi/2)/sqrt(33). - Vaclav Kotesovec, Dec 31 2022

A056115 a(n) = n*(n+11)/2.

Original entry on oeis.org

0, 6, 13, 21, 30, 40, 51, 63, 76, 90, 105, 121, 138, 156, 175, 195, 216, 238, 261, 285, 310, 336, 363, 391, 420, 450, 481, 513, 546, 580, 615, 651, 688, 726, 765, 805, 846, 888, 931, 975, 1020, 1066, 1113, 1161, 1210, 1260, 1311, 1363, 1416, 1470, 1525

Offset: 0

Barry E. Williams, Jul 04 2000

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

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000096, A001477, A055999, A056000, A056119.

Third column of Pascal (1, 6) triangle A096956.

List([0..50], n-> n*(n+11)/2 ); # G. C. Greubel, Jan 18 2020

[n*(n+11)/2: n in [0..50]]; // G. C. Greubel, Jan 18 2020

((2*Range[0,50]+11)^2 -11^2)/8 (* G. C. Greubel, Jan 18 2020 *)

a(n)=n*(n+11)/2; \\ Joerg Arndt, Oct 25 2014

[n*(n+11)/2 for n in (0..50)] # G. C. Greubel, Jan 18 2020

G.f.: x*(6-5*x)/(1-x)^3.
a(n) = A000096(n) + 4*A001477(n) = A056000(n) + A001477(n) = A056119(n) - A001477(n). - Zerinvary Lajos, Oct 01 2006
a(n) = A126890(n,5) for n>4. - Reinhard Zumkeller, Dec 30 2006
Equals A119412/2. - Zerinvary Lajos, Feb 12 2007
If we define f(n,i,a) = Sum_{k=0..n-i} ( binomial(n,k)*stirling1(n-k,i) *Product_{j=0..k-1} (-a-j) ), then a(n) = -f(n,n-1,6), for n>=1. - Milan Janjic, Dec 20 2008
a(n) = a(n-1) + n + 5 (with a(0)=0). - Vincenzo Librandi, Aug 07 2010
Sum_{n>=1} 1/a(n) = 83711/152460. - R. J. Mathar, Jul 14 2012
a(n) = 6*n - floor(n/2) + floor(n^2/2). - Wesley Ivan Hurt, Jun 15 2013
E.g.f.: x*(12 + x)*exp(x)/2. - G. C. Greubel, Jan 18 2020
Sum_{n>=1} (-1)^(n+1)/a(n) = 4*log(2)/11 - 20417/152460. - Amiram Eldar, Jan 10 2021

A139570 a(n) = 2n(n+3).

Original entry on oeis.org

0, 8, 20, 36, 56, 80, 108, 140, 176, 216, 260, 308, 360, 416, 476, 540, 608, 680, 756, 836, 920, 1008, 1100, 1196, 1296, 1400, 1508, 1620, 1736, 1856, 1980, 2108, 2240, 2376, 2516, 2660, 2808, 2960, 3116, 3276, 3440, 3608, 3780, 3956, 4136, 4320, 4508, 4700, 4896

Offset: 0

Omar E. Pol, May 19 2008

Numbers n such that 2*n + 9 is a square. - Vincenzo Librandi, Nov 24 2010

a(n) appears also as the fourth member of the quartet [p0(n), p1(n), p2(n), a(n)] of the square of [n, n+1, n+2, n+3] in the Clifford algebra Cl_2 for n >= 0. p0(n) = -A147973(n+3), p1(n) = A046092(n), and p2(n) = A054000(n+1). See a comment on A147973, also with a reference. - Wolfdieter Lang, Oct 15 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000096, A001105, A020739, A022998, A028552, A046092, A054000, A067728, A147973.

CoefficientList[Series[4 x (2 - x)/(1 - x)^3, {x, 0, 40}], x] (* Vincenzo Librandi, May 23 2014 *)

a(n)=2*n*(n+3) \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 2*A028552(n) = 2*n^2 + 6*n = n*(2*n+6).
a(n) = a(n-1) + 4*n + 4 (with a(0)=0). - Vincenzo Librandi, Nov 24 2010
From Paul Curtz, Mar 27 2011: (Start)
a(n) = A022998(n)*A022998(n+3).
a(n) = 4*A000096(n). (End)
G.f.: 4*x*(2 - x)/(1 - x)^3. - Arkadiusz Wesolowski, Dec 31 2011
From Amiram Eldar, Dec 23 2022: (Start)
Sum_{n>=1} 1/a(n) = 11/36.
Sum_{n>=1} (-1)^(n+1)/a(n) = log(2)/3 - 5/36. (End)
From Elmo R. Oliveira, Nov 16 2024: (Start)
E.g.f.: 2*exp(x)*x*(4 + x).
a(n) = n*A020739(n).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

A227819 Number T(n,k) of n-node rooted identity trees of height k; triangle T(n,k), n>=1, 0<=k<=n-1, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 2, 1, 0, 0, 0, 2, 3, 1, 0, 0, 0, 2, 5, 4, 1, 0, 0, 0, 2, 8, 9, 5, 1, 0, 0, 0, 1, 12, 18, 14, 6, 1, 0, 0, 0, 1, 17, 34, 33, 20, 7, 1, 0, 0, 0, 1, 23, 61, 72, 54, 27, 8, 1, 0, 0, 0, 0, 32, 108, 149, 132, 82, 35, 9, 1, 0, 0, 0, 0, 41, 187, 301, 303, 221, 118, 44, 10, 1

Offset: 1

Alois P. Heinz, Jul 31 2013

			:   T(6,4) = 3              :  T(11,3) = 1  :
:     o       o       o     :        o      :
:    / \      |       |     :      /( )\    :
:   o   o     o       o     :     o o o o   :
:   |        / \      |     :    /| | |     :
:   o       o   o     o     :   o o o o     :
:   |       |        / \    :   |   |       :
:   o       o       o   o   :   o   o       :
:   |       |       |       :               :
:   o       o       o       :               :
Triangle T(n,k) begins:
  1;
  0, 1;
  0, 0, 1;
  0, 0, 1, 1;
  0, 0, 0, 2,  1;
  0, 0, 0, 2,  3,   1;
  0, 0, 0, 2,  5,   4,   1;
  0, 0, 0, 2,  8,   9,   5,   1;
  0, 0, 0, 1, 12,  18,  14,   6,  1;
  0, 0, 0, 1, 17,  34,  33,  20,  7,  1;
  0, 0, 0, 1, 23,  61,  72,  54, 27,  8, 1;
  0, 0, 0, 0, 32, 108, 149, 132, 82, 35, 9, 1;

Alois P. Heinz, Rows n = 1..141, flattened

Columns k=4-10 give: A038088, A038089, A038090, A038091, A038092, A229403, A229404.
Row sums give: A004111.
Column sums give: A038081.
Largest n with T(n,k)>0 is A038093(k).
Main diagonal and lower diagonals give (offsets may differ): A000012, A001477, A000096, A166830.
T(2n,n) gives A245090.
T(2n+1,n) gives A245091.
Cf. A034781.

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1 or k<1, 0,
      add(binomial(b((i-1)$2, k-1), j)*b(n-i*j, i-1, k), j=0..n/i)))
    end:
T:= (n, k)-> b((n-1)$2, k) -`if`(k=0, 0, b((n-1)$2, k-1)):
seq(seq(T(n, k), k=0..n-1), n=1..15);

Drop[Transpose[Map[PadRight[#,15]&,Table[f[n_]:=Nest[ CoefficientList[ Series[ Product[(1+x^i)^#[[i]],{i,1,Length[#]}],{x,0,15}],x]&,{1},n]; f[m]-PadRight[f[m-1],Length[f[m]]],{m,1,15}]]],1]//Grid (* Geoffrey Critzer, Aug 01 2013 *)

A006503 a(n) = n(n+1)(n+8)/6.

Original entry on oeis.org

0, 3, 10, 22, 40, 65, 98, 140, 192, 255, 330, 418, 520, 637, 770, 920, 1088, 1275, 1482, 1710, 1960, 2233, 2530, 2852, 3200, 3575, 3978, 4410, 4872, 5365, 5890, 6448, 7040, 7667, 8330, 9030, 9768, 10545, 11362, 12220, 13120, 14063, 15050, 16082, 17160, 18285

Offset: 0

N. J. A. Sloane

If Y is a 3-subset of an n-set X then, for n>=4, a(n-4) is the number of 3-subsets of X having at most one element in common with Y. - Milan Janjic, Nov 23 2007

The coefficient of x^3 in (1-x-x^2)^{-n} is the coefficient of x^3 in (1+x+2x^2+3x^3)^n. Using the multinomial theorem one then finds that a(n)=n(n+1)(n+8)/3!. - Sergio Falcon, May 22 2008

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

G. C. Greubel, Table of n, a(n) for n = 0..5000
Margaret Bayer, Mark Denker, Marija Jelić Milutinović, Rowan Rowlands, Sheila Sundaram, and Lei Xue, Topology of Cut Complexes of Graphs, arXiv:2304.13675 [math.CO], 2023.
G. E. Bergum and V. E. Hoggatt, Jr., Numerator polynomial coefficient array for the convolved Fibonacci sequence, Fib. Quart., 14 (1976), 43-44. (Annotated scanned copy)
G. E. Bergum and V. E. Hoggatt, Jr., Numerator polynomial coefficient array for the convolved Fibonacci sequence, Fib. Quart., 14 (1976), 43-48.
Milan Janjić, Hessenberg Matrices and Integer Sequences, J. Int. Seq. 13 (2010) # 10.7.8, section 3.
P. Moree, Convoluted convolved Fibonacci numbers, arXiv:math/0311205 [math.CO], 2003.
P. Moree, Convoluted Convolved Fibonacci Numbers, Journal of Integer Sequences, Vol. 7 (2004), Article 04.2.2.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

a(n) = A095660(n+2, 3): fourth column of (1, 3)-Pascal triangle.
Cf. A000027, A000096, A006504.
Cf. A000292, A002378.
Row n=3 of A144064.

A006503:=-(-3+2*z)/(z-1)**4; # [Simon Plouffe in his 1992 dissertation.]

Clear["Global`*"] a[n_] := n(n + 1)(n + 8)/3! Do[Print[n, " ", a[n]], {n, 1, 25}] (* Sergio Falcon, May 22 2008 *)
Table[n(n+1)(n+8)/6,{n,0,50}] (* or *) LinearRecurrence[{4,-6,4,-1},{0,3,10,22},50] (* Harvey P. Dale, Jan 27 2016 *)

x='x+O('x^50); concat([0], Vec(x*(3-2*x)/(1-x)^4)) \\ G. C. Greubel, May 11 2017

a(n) = n*(n+1)*(n+8)/6.
G.f.: x*(3-2*x)/(1-x)^4.
a(n) = A000292(n) + A002378(n). - Reinhard Zumkeller, Sep 24 2008
a(n) = 4*a(n-1)-6*a(n-2)+ 4*a(n-3)- a(n-4) with a(0)=0, a(1)=3, a(2)=10, a(3)=22. - Harvey P. Dale, Jan 27 2016

Better description from Jeffrey Shallit, Aug 1995

A056121 a(n) = n*(n + 15)/2.

Original entry on oeis.org

0, 8, 17, 27, 38, 50, 63, 77, 92, 108, 125, 143, 162, 182, 203, 225, 248, 272, 297, 323, 350, 378, 407, 437, 468, 500, 533, 567, 602, 638, 675, 713, 752, 792, 833, 875, 918, 962, 1007, 1053, 1100, 1148, 1197, 1247, 1298, 1350, 1403, 1457, 1512, 1568, 1625

Offset: 0

Barry E. Williams, Jul 06 2000

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000096, A001477, A056000, A056119, A056126.

List([0..60], n-> n*(n+15)/2 ); # G. C. Greubel, Jan 18 2020

[n*(n+15)/2: n in [0..60]]; // G. C. Greubel, Jan 18 2020

Maple
```
a:=n->n*(n+15)/2: seq(a(n),n=0..60);
```

Table[n*(n + 15)/2, {n, 0, 100}] (* Paolo Xausa, Aug 02 2024 *)

a(n)=n*(n+15)/2 \\ Charles R Greathouse IV, Sep 24 2015

[n*(n+15)/2 for n in (0..60)] # G. C. Greubel, Jan 18 2020

G.f.: x*(8-7*x)/(1-x)^3.
a(n) = A000096(n) + 6*n = A056119(n) + n = A056126(n) - n. - Zerinvary Lajos, Oct 01 2006
a(n-15) = binomial(n,2) - 7*n. - Zerinvary Lajos, Nov 26 2006
a(n) = A126890(n,7) for n>6. - Reinhard Zumkeller, Dec 30 2006
Let f(n,i,a) = Sum_{k=0..n-i} binomial(n,k)*Stirling1(n-k,i)*Product_{j=0..k-1} (-a-j), then a(n) = -f(n,n-1,8), for n>=1. - Milan Janjic, Dec 20 2008
a(n) = a(n-1)+ n + 7 (with a(0)=0). - Vincenzo Librandi, Aug 07 2010
Sum_{n>=1} 1/a(n) = 1195757/2702700 via A132760. - R. J. Mathar, Jul 14 2012
a(n) = 8*n - floor(n/2) + floor(n^2/2). - Wesley Ivan Hurt, Jun 15 2013
E.g.f.: x*(16 + x)*exp(x)/2. - G. C. Greubel, Jan 18 2020
Sum_{n>=1} (-1)^(n+1)/a(n) = 4*log(2)/15 - 52279/540540. - Amiram Eldar, Jan 10 2021

More terms from James Sellers, Jul 07 2000

cp's OEIS Frontend

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A046691 a(n) = (n^2 + 5*n - 2)/2.

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A126890 Triangle read by rows: T(n,k) = n(n+2k+1)/2, 0 <= k <= n.

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A006503 a(n) = n(n+1)(n+8)/6.