A231599 -id:A231599 - cp's OEIS frontend

A269298 Central nonzero values of A231599.

1, 2, 2, 4, 6, 8, 16, 28, 50, 100, 196, 388, 786, 1600, 3280, 6780, 14060, 29280, 61232, 128414, 270084, 569514, 1203564, 2548770, 5407754, 11493312, 24465960, 52157508, 111342192, 237985596, 509275390, 1091017632, 2339687834, 5022312654, 10790564790

Offset: 0

Tilman Piesk, Feb 21 2016

nonn

Rows of A231599 whose row number is divisible by four have positive central values. a(n) is the central value of row 4n. They are also the maximal value of that row, so a(n) = A086376(4n).

a(1) = a(2) = 2. Apart from that the sequence is strictly increasing.

			For n = 5, A231599( 4n, A000217(4n)/2 ) = A231599(20, 105) = 8, so a(5)=8.
For n = 5, A086376(4n) = A086376(20) = 8, so a(5)=8.

Alois P. Heinz, Table of n, a(n) for n = 0..250 (first 101 trms from Tilman Piesk)
Dorin Andrica and Ovidiu Bagdasar, On some results concerning the polygonal polynomials, Carpathian Journal of Mathematics (2019) Vol. 35, No. 1, 1-11.
Tilman Piesk, A231599 as a centered table (first 11 values visible)

Cf. A231599, A086376.

a(n) = A231599( 4n, A000217(4n)/2 ) = A086376(4n).

A053632 Irregular triangle read by rows giving coefficients in expansion of Product_{k=1..n} (1 + x^k).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Mar 22 2000

Or, triangle T(n,k) read by rows, giving number of subsets of {1,2,...,n} with sum k. - Roger CUCULIERE (cuculier(AT)imaginet.fr), Nov 19 2000
Row n consists of A000124(n) terms. These are also the successive vectors (their nonzero elements) when one starts with the infinite vector (of zeros) with 1 inserted somewhere and then shifts it one step (right or left) and adds to the original, then shifts the result two steps and adds, three steps and adds, etc. - Antti Karttunen, Feb 13 2002
T(n,k) = number of partitions of k into distinct parts <= n. Triangle of distribution of Wilcoxon's signed rank statistic. - Mitch Harris, Mar 23 2006
T(n,k) = number of binary words of length n in which the sum of the positions of the 0's is k. Example: T(4,5)=2 because we have 0110 (sum of the positions of the 0's is 1+4=5) and 1001 (sum of the positions of the 0's is 2+3=5). - Emeric Deutsch, Jul 23 2006
A fair coin is flipped n times. You receive i dollars for a "success" on the i-th flip, 1<=i<=n. T(n,k)/2^n is the probability that you will receive exactly k dollars. Your expectation is n(n+1)/4 dollars. - Geoffrey Critzer, May 16 2010
From Gus Wiseman, Jan 02 2023: (Start)
With offset 1, also the number of integer compositions of n whose partial sums add up to k for k = n..n(n+1)/2. For example, row n = 6 counts the following compositions:
  6 15 24 33  42  51  141  231  321  411  1311 2211  3111  12111 21111 111111
          114 123 132 222  312  1131 1221 2121 11121 11211
                  213 1113 1122 1212 2112 1111
(End)

			Triangle begins:
  1;
  1, 1;
  1, 1, 1, 1;
  1, 1, 1, 2, 1, 1, 1;
  1, 1, 1, 2, 2, 2, 2, 2, 1, 1, 1;
  1, 1, 1, 2, 2, 3, 3, 3, 3, 3, 3, 2, 2, 1, 1, 1;
  1, 1, 1, 2, 2, 3, 4, 4, 4, 5, 5, 5, 5, 4, 4, 4, 3, 2, 2, 1, 1, 1;
  ...
Row n = 4 counts the following binary words, where k = sum of positions of zeros:
  1111  0111  1011  0011  0101  0110  0001  0010  0100  1000  0000
                    1101  1110  1001  1010  1100
Row n = 5 counts the following strict partitions of k with all parts <= n (0 is the empty partition):
  0  1  2  3  4  5  42  43  53  54  532  542  543  5431 5432 54321
           21 31 32 51  52  431 432 541  5321 5421
                 41 321 421 521 531 4321

A. V. Yurkin, New binomial and new view on light theory, (book), 2013, 78 pages, no publisher listed.

Alois P. Heinz, Rows n = 0..40, flattened
S. R. Finch, Signum equations and extremal coefficients.
Steven R. Finch, Signum equations and extremal coefficients, February 7, 2009. [Cached copy, with permission of the author]
FindStat - Combinatorial Statistic Finder, The major index of an integer composition
Alexander Rosa and Štefan Znám, A combinatorial problem in the theory of congruences. (Russian), Mat.-Fys. Casopis Sloven. Akad. Vied 15 1965 49-59. [Annotated scanned copy] See Table 1.
F. Wilcoxon, Individual Comparisons by Ranking Methods, Biometrics Bulletin, v. 1, no. 6 (1945), pp. 80-83.
A. V. Yurkin, On similarity of systems of geometrical and arithmetic triangles, in Mathematics, Computing, Education Conference XIX, 2012.
A. V. Yurkin, New view on the diffraction discovered by Grimaldi and Gaussian beams, arXiv preprint arXiv:1302.6287 [physics.optics], 2013.
A. V. Yurkin, About the evident description of distribution of beams and "wavy geometrical trajectories" in long thin pipes, 2014 (original in Russian).
A. V. Yurkin, Symmetric triangle of Pascal and non-linear arithmetic parallelepiped, Book Manuscript, Research Gate 2015.

Cf. A053633, A068009.
Rows reduced modulo 2 and interpreted as binary numbers: A068052, A068053. Rows converge towards A000009.
Row sums give A000079.
Cf. A001788, A028362.
Cf. A285101 (multiplicative encoding of each row), A285103 (number of odd terms on row n), A285105 (number of even terms).
Row lengths are A000124.
A reciprocal version is (A033999, A219977, A291983, A291984, A291985, ...).
A negative version is A231599.
A version for partitions is A358194, reversed partitions A264034.
Cf. A025591, A029931, A063865, A152947, A318283, A359042.

with(gfun,seriestolist); map(op,[seq(seriestolist(series(mul(1+(z^i), i=1..n),z,binomial(n+1,2)+1)), n=0..10)]); # Antti Karttunen, Feb 13 2002
# second Maple program:
g:= proc(n) g(n):= `if`(n=0, 1, expand(g(n-1)*(1+x^n))) end:
T:= n-> seq(coeff(g(n), x, k), k=0..degree(g(n))):
seq(T(n), n=0..10);  # Alois P. Heinz, Nov 19 2012

Table[CoefficientList[ Series[Product[(1 + t^i), {i, 1, n}], {t, 0, 100}], t], {n, 0, 8}] // Grid (* Geoffrey Critzer, May 16 2010 *)

From Mitch Harris, Mar 23 2006: (Start)
T(n,k) = T(n-1, k) + T(n-1, k-n), T(0,0)=1, T(0,k) = 0, T(n,k) = 0 if k < 0 or k > (n+1 choose 2).
G.f.: (1+x)*(1+x^2)*...*(1+x^n). (End)
Sum_{k>=0} k * T(n,k) = A001788(n). - Alois P. Heinz, Feb 09 2017
max_{k>=0} T(n,k) = A025591(n). - Alois P. Heinz, Jan 20 2023

A026807 Triangular array T read by rows: T(n,k) = number of partitions of n in which every part is >=k, for k=1,2,...,n.

Original entry on oeis.org

1, 2, 1, 3, 1, 1, 5, 2, 1, 1, 7, 2, 1, 1, 1, 11, 4, 2, 1, 1, 1, 15, 4, 2, 1, 1, 1, 1, 22, 7, 3, 2, 1, 1, 1, 1, 30, 8, 4, 2, 1, 1, 1, 1, 1, 42, 12, 5, 3, 2, 1, 1, 1, 1, 1, 56, 14, 6, 3, 2, 1, 1, 1, 1, 1, 1, 77, 21, 9, 5, 3, 2, 1, 1, 1, 1, 1, 1, 101, 24, 10, 5, 3, 2, 1, 1, 1, 1, 1, 1, 1, 135, 34, 13

Offset: 1

Clark Kimberling

T(n,g) is also the number of not necessarily connected 2-regular graphs with girth at least g: the part i corresponds to the i-cycle; addition of integers corresponds to disconnected union of cycles. - Jason Kimberley, Feb 05 2012

			Sum_{k>=1} y^k*(-1+1/Product_{i>=0} (1-x^(k+i))) = y*x+(2*y+y^2)*x^2+(3*y+y^2+y^3)*x^3+(5*y+2*y^2+y^3+y^4)*x^4+(7*y+2*y^2+y^3+y^4+y^5)*x^5+...
Triangle starts:  - _Jason Kimberley_, Feb 05 2012
1;
2, 1;
3, 1, 1;
5, 2, 1, 1;
7, 2, 1, 1, 1;
11, 4, 2, 1, 1, 1;
15, 4, 2, 1, 1, 1, 1;
22, 7, 3, 2, 1, 1, 1, 1;
30, 8, 4, 2, 1, 1, 1, 1, 1;
42, 12, 5, 3, 2, 1, 1, 1, 1, 1;
56, 14, 6, 3, 2, 1, 1, 1, 1, 1, 1;
77, 21, 9, 5, 3, 2, 1, 1, 1, 1, 1, 1;
101, 24, 10, 5, 3, 2, 1, 1, 1, 1, 1, 1, 1;
From _Tilman Piesk_, Feb 20 2016: (Start)
n = 12, k = 4, t = A000217(k-1) = 6
vp = A000041(n..n-t) = A000041(12..6) = (77, 56, 42, 30, 22, 15, 11)
vc = A231599(k-1, 0..t) = A231599(3, 0..6) = (1,-1,-1, 0, 1, 1,-1)
T(12, 4) = vp * transpose(vc) = 77-56-42+22+15-11 = 5
(End)

Alois P. Heinz, Rows n = 1..141, flattened
Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth at least g
Tilman Piesk, Table for n = 1..30, table for n = 2..150 without values 1, illustrations of columns n = 2, 3, 4, 5, 6, 7, 8

Row sums give A046746.
Cf. A026835.
Cf. A026794.
Cf. A231599.
Not necessarily connected 2-regular graphs with girth at least g [partitions into parts >= g]: this sequence (triangle); columns of this sequence: A000041 (g=1 -- multigraphs with loops allowed), A002865 (g=2 -- multigraphs with loops forbidden), A008483 (g=3), A008484 (g=4), A185325(g=5), A185326 (g=6), A185327 (g=7), A185328 (g=8), A185329 (g=9). For g >= 3, girth at least g implies no loops or parallel edges. - Jason Kimberley, Feb 05 2012
Not necessarily connected 2-regular simple graphs with girth exactly g [partitions with smallest part g]: A026794 (triangle); chosen g: A002865 (g=2), A026796 (g=3), A026797 (g=4), A026798 (g=5), A026799 (g=6), A026800(g=7), A026801 (g=8), A026802 (g=9), A026803 (g=10). - Jason Kimberley, Feb 05 2012
Cf. A002260.

import Data.List (tails)
a026807 n k = a026807_tabl !! (n-1) !! (k-1)
a026807_row n = a026807_tabl !! (n-1)
a026807_tabl = map
   (\row -> map (p $ last row) $ init $ tails row) a002260_tabl
   where p 0  _ = 1
         p _ [] = 0
         p m ks'@(k:ks) = if m < k then 0 else p (m - k) ks' + p m ks
-- Reinhard Zumkeller, Dec 01 2012

T:= proc(n, k) option remember;
      `if`(k<1 or k>n, 0, `if`(n=k, 1, T(n, k+1) +T(n-k, k)))
    end:
seq(seq(T(n, k), k=1..n), n=1..14); # Alois P. Heinz, Mar 28 2012

T[n_, k_] := T[n, k] = If[ k<1 || k>n, 0, If[n == k, 1, T[n, k+1] + T[n-k, k]]]; Table [Table[ T[n, k], {k, 1, n}], {n, 1, 14}] // Flatten (* Jean-François Alcover, Jan 28 2015, after Alois P. Heinz *)

from see_there import a231599_row  # A231599
from sympy.ntheory import npartitions  # A000041
def a026807(n, k):
    if k > n:
        return 0
    elif k > n/2:
        return 1
    else:
        vc = a231599_row(k-1)
        t = len(vc)
        vp_range = range(n-t, n+1)
        vp_range = vp_range[::-1]  # reverse
        r = 0
        for i in range(0, t):
            r += vc[i] * npartitions(vp_range[i])
        return r
# Tilman Piesk, Feb 21 2016

T(n,1)=A000041(n), T(n,2)=A002865(n) for n>1, T(n,3)=A008483(n) for n>2, T(n,4)=A008484(n) for n>3.
G.f.: Sum_{k>=1} y^k*(-1+1/Product_{i>=0} (1-x^(k+i))). - Vladeta Jovovic, Jun 22 2003
T(n, k) = T(n, k+1) + T(n-k, k), T(n, k) = 1 if n/2 < k <= n. - Franklin T. Adams-Watters, Jan 24 2005; Tilman Piesk, Feb 20 2016
T(n, k) = A000041(n..n-t) * transpose(A231599(k-1, 0..t)) with t = A000217(k-1). - Tilman Piesk, Feb 20 2016
Equals A026794 * A000012 as infinite lower triangular matrices. - Gary W. Adamson, Jan 31 2008

A282634 Recursive 2-parameter sequence allowing the Ramanujan's sum calculation.

Original entry on oeis.org

1, 1, -1, 2, -1, -1, 2, 0, -2, 0, 4, -1, -1, -1, -1, 2, 1, -1, -2, -1, 1, 6, -1, -1, -1, -1, -1, -1, 4, 0, 0, 0, -4, 0, 0, 0, 6, 0, 0, -3, 0, 0, -3, 0, 0, 4, 1, -1, 1, -1, -4, -1, 1, -1, 1, 10, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 4, 0, 2, 0, -2, 0, -4, 0

Offset: 1

Gevorg Hmayakyan, Feb 20 2017

a(n,0) = phi(n), where phi(n) is Euler's totient function A000010(n).

a(n,1) = mu(n), where mu(n) is the Möbius function A008683(n).

			The few first rows follow:            c_n(t)
  t   0   1   2   3   4   5   6     |  t   1   2   3   4   5   6   7
n                                   |n
1     1;                            |1     1;
2     1, -1;                        |2    -1,  1;
3     2, -1, -1;                    |3    -1, -1,  2;
4     2,  0, -2,  0;                |4     0, -2,  0,  2;
5     4, -1, -1, -1, -1;            |5    -1, -1, -1, -1,  4;
6     2,  1, -1, -2, -1,  1;        |6     1, -1, -2, -1,  1,  2;
7     6, -1, -1, -1, -1, -1, -1;    |7    -1, -1, -1, -1, -1, -1,  6;
      ...                           |     ...
[Edited by _Seiichi Manyama_, Mar 05 2018]

Seiichi Manyama, Rows n=1..140 of triangle, flattened
Gevorg Hmayakyan, On The Moebius and Euler Totient Functions Calculation.
Charles A. Nicol, On Restricted Partitions and a Generalization Of The Euler Totient and The Moebius Function, PNAS 39(9) (1953), 963-968.

Cf. A000010 (phi(n)), A008683 (mu(n)), A054532, A054533, A054534, A054535, A231599.

b[n_, m_] := b[n, m] = If[n > 1, b[n - 1, m] - b[n - 1, m - n + 1], 0]
b[1, m_] := b[1, m] = If[m == 0, 1, 0]
nt[n_, t_] := Round[(n - 1)/2 - t/n]
a[n_, t_] := Sum[b[n, k*n + t], {k, 0, nt[n, t]}]
Flatten[Table[Table[a[n, m], {m, 0, n - 1}], {n, 1, 20}]]

a(n,t) = Sum(b(n, k*n + t), k=0..N(n, t)), where b(n,k) = A231599(n-1,k) and N(n,t) = [(n - 1)/2 - t/n].
a(n,t) = c_n(t) for t >= 1, where c_n(t) is a Ramanujan's sum A054533.
a(n,t) = a(n,-t)
From Seiichi Manyama, Mar 05 2018: (Start)
a(n,t) = c_n(n-t) = Sum_{d | gcd(n,n-t)} d*mu(n/d) for 0 <= t <= n-1.
So a(n,t) = Sum_{d | gcd(n,t)} d*mu(n/d) for 1 <= t <= n-1. (End)

A303992 Triangular array T(n,k) giving coefficients in expansion of Product_{j=1..n} (1-x^j)^3.

Original entry on oeis.org

1, 1, -3, 3, -1, 1, -3, 0, 8, -6, -6, 8, 0, -3, 1, 1, -3, 0, 5, 3, -6, -13, 9, 15, 0, -15, -9, 13, 6, -3, -5, 0, 3, -1, 1, -3, 0, 5, 0, 3, -13, -6, 9, 9, 24, -21, -24, -9, 3, 44, 3, -9, -24, -21, 24, 9, 9, -6, -13, 3, 0, 5, 0, -3, 1

Offset: 0

Seiichi Manyama, May 04 2018

			Irregular triangle starts:
n\k| 0   1  2   3   4   5    6  7   8  9   10  11  12 13  14  15 16 17  18
---+-----------------------------------------------------------------------
0  | 1;
1  | 1, -3, 3, -1;
2  | 1, -3, 0,  8, -6, -6,   8, 0, -3, 1;
3  | 1, -3, 0,  5,  3, -6, -13, 9, 15, 0, -15, -9, 13, 6, -3, -5, 0, 3, -1;

Seiichi Manyama, Rows n = 0..26, flattened

Cf. A010816, A231599, A304080.

T(n, k) = polcoef(prod(j=1, n, (1-x^j)^3), k);
tabf(nn) = for(n=0, nn, for(k=0, 3*n*(n+1)/2, print1(T(n, k), ", ")); print)

A304080 Triangular array T(n,k) giving coefficients in expansion of Product_{j=1..n} (1-x^j)^2.

Original entry on oeis.org

1, 1, -2, 1, 1, -2, -1, 4, -1, -2, 1, 1, -2, -1, 2, 3, 0, -6, 0, 3, 2, -1, -2, 1, 1, -2, -1, 2, 1, 4, -4, -4, -2, 0, 10, 0, -2, -4, -4, 4, 1, 2, -1, -2, 1, 1, -2, -1, 2, 1, 2, 0, -2, -6, -2, 3, 6, 5, 2, -3, -12, -3, 2, 5, 6, 3, -2, -6, -2, 0, 2, 1, 2, -1, -2, 1

Offset: 0

Seiichi Manyama, May 06 2018

			Irregular triangle starts:
n\k| 0   1   2  3   4   5   6   7   8  9  10  11  12  13  14 15 16 17  18  19 20
---+-----------------------------------------------------------------------------
0  | 1;
1  | 1, -2,  1;
2  | 1, -2, -1, 4, -1, -2,  1;
3  | 1, -2, -1, 2,  3,  0, -6,  0,  3, 2, -1, -2,  1;
4  | 1, -2, -1, 2,  1,  4, -4, -4, -2, 0, 10,  0, -2, -4, -4, 4, 1, 2, -1, -2, 1;

Seiichi Manyama, Rows n = 0..30, flattened

Cf. A002107, A231599, A303992.

T(n, k) = polcoef(prod(j=1, n, (1-x^j)^2), k);
tabf(nn) = for(n=0, nn, for(k=0, n*(n+1), print1(T(n, k), ", ")); print)

A333290 Irregular triangle read by rows: coefficients b_{r,j} (r>=1, j>=0) arising from an expansion of the partition function.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Mar 20 2020

Is this (apart from offset) the same as A231599? - R. J. Mathar, Mar 21 2020

			Triangle begins:
1,
1,-1,
1,-1,-1,1,
1,-1,-1,0,1,1,-1,
1,-1,-1,0,0,2,0,0,-1,-1,1,
1,-1,-1,0,0,1,1,1,-1,-1,-1,0,0,1,1,-1,
...

Mircea Merca and Maxie D. Schmidt, The partition function p(n) in terms of the classical Möbius function, Ramanujan J (2019) 49:87-96.

Cf. A333289.

A333290 := proc(r,j)
    if r < 1 then
        0 ;
    elif r = 1 then
        if j= 0 then
            1;
        else
            0 ;
        end if;
    elif j < r-1 then
        procname(r-1,j) ;
    else
        procname(r-1,j) -procname(r-1,j-r+1) ;;
    end if;
end proc: # R. J. Mathar, Mar 21 2020

b[r_, j_] := b[r, j] = Which[r < 1, 0, r == 1, If[j == 0, 1, 0], j < r-1, b[r-1, j], True, b[r-1, j] - b[r-1, j-r+1]];
Table[b[r, j], {r, 1, 9}, {j, 0, r(r-1)/2}] // Flatten (* Jean-François Alcover, Apr 29 2023, after R. J. Mathar *)

A061553 Sum of absolute values of coefficients of expansion of (1-x)(1-x^2)(1-x^3)...(1-x^n).

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 20, 28, 36, 44, 54, 72, 92, 104, 138, 176, 212, 268, 332, 416, 508, 628, 776, 968, 1192, 1480, 1836, 2288, 2812, 3472, 4292, 5312, 6572, 8120, 10028, 12388, 15300, 18860, 23276, 28740, 35468, 43732, 53954, 66540, 82016, 101044

Offset: 0

Steffen Eckmann (steffen.eckmann(AT)eon.com), May 17 2001

nonn

a(n) >= A160089(n) with equality only for n=0. - Michel Marcus, Jun 12 2013

			a(1) = 1+1 = 2; a(4) = Length(P(4,x)) = Length(1 - x - x^2 + 2x^5 - x^8 - x^9 + x^10) = 1+1+1+2+1+1+1 = 8

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A160089, A231599.

a(n) = {pol = prod(i=1, n, 1-x^i); return (sum(i=0, poldegree(pol), abs(polcoeff(pol, i))));} \\ Michel Marcus, Jun 12 2013

a(n) := |c(n, 0)| + |c(n, 1)| + ... + |c(n, n(n+1)/2)| where c(n, j) are the coefficients of the polynomial P(n, x) := (1-x)(1-x^2)(1-x^3)...(1-x^n)

a(0)=1 prepended by Seiichi Manyama, May 03 2018

A282283 Recursive 2-parameter sequence allowing calculation of the Euler Totient function.

Original entry on oeis.org

0, 1, -1, 1, 2, -4, 2, -4, 10, -6, -2, 2, 6, -16, 10, 4, -6, 8, -10, 4, -10, 28, -18, -8, 10, -10, 10, -2, 8, -10, 0, 2, 12, -34, 22, 10, -12, 12, -22, 30, -30, 6, 10, -10, 8, 0, 6, -14, 6, -18, 52, -34, -16, 18, -18, 34, -36, 20, 10, -6, -2, 4, -28, 18, 8

Offset: 0

Gevorg Hmayakyan, Feb 11 2017

The a(n,m) forms a table where each row has (n*(n-3)+4)/2 = A152947(n) elements.
The index of the first row is n=1 and the index of the first column is m=0.
The right diagonal a(n, A152947(n)) = A000010(n), Euler Totient function.

			The first few rows are:
0, 1;
-1, 1;
2, -4, 2;
-4, 10, -6, -2, 2;
6, -16, 10, 4, -6, 8, -10, 4;
-10, 28, -18, -8, 10, -10, 10, -2, 8, -10, 0, 2;
12, -34, 22, 10, -12, 12, -22, 30, -30, 6, 10, -10, 8, 0, 6, -14, 6;

Cf. A000010, A152947, A231599.

U[n_, m_] := U[n, m] = If[n > 1, U[n - 1, n*(n - 1)/2 - m]*(-1)^n - U[n - 1, m], 0]
U[1, m_] := U[1, m] = If[m == 0, 1, 0]
Q[n_, m_] := U[n, m - 2] - 2*U[n, m - 1] + U[n, m]
nu[n_]:=(n-1)*n/2+2-n
a[n_, m_] := a[n, m] = If[(m < 0) || (nu[n] < m), 0, a[n - 1, m - n + 1] - a[n - 1, m] - a[n - 1, nu[n - 1]]*Q[n - 1, m]]
a[1, m_] := a[1, m] = If[m == 1, 1, 0]
Table[Table[a[n, m], {m, 0, nu[n]}], {n, 1, 20}]
Table[a[n, nu[n]], {n, 1, 50}]

nu(n) = (n*(n-3)+4)/2
Q(n,m) = 2*A231599(n,m-1)-A231599(n,m-2)-A231599(n,m)
a(n, m) = a(n - 1, m - n + 1) - a(n - 1, m) - a(n - 1, nu(n - 1))*Q(n - 1, m) if (m < 0) or (nu(n) < m)
a(1,m)=1 if m=1 and 0 otherwise.
a(n,nu(n))= A000010(n)

A301701 a(n) is the smallest positive integer m, with the property that n appears as a coefficient in the polynomial P_m(x) = (x-1)(x^2-1)...(x^m-1).

Original entry on oeis.org

3, 1, 4, 10, 12, 17, 16, 19, 20, 22, 22, 23, 24, 25, 25, 25, 24, 26, 26, 28, 27, 27, 29, 28, 28, 29, 29, 30, 28, 29, 30, 30, 30, 30, 30, 31, 31, 31, 31, 31, 31, 31, 32, 33, 33, 33, 32, 33, 32, 33, 32, 33, 33, 36, 35, 33, 33, 36, 34, 34, 37, 35, 34, 37, 35, 34, 34, 35, 35, 35, 35

Offset: 0

Ovidiu Bagdasar, Mar 25 2018

nonn

We conjecture that all integers appear as a coefficient of a polynomial P_m(x).
This property is known to hold for the cyclotomic polynomials.
The conjecture holds for the first 10^5 positive integers, with a maximum on those integers of a(99852) = 1921. - David A. Corneth, Apr 08 2018

			We have:
P_1(x) = x-1, hence a(1)=1.
P_2(x) = (x-1)*(x^2-1) = x^3-x^2-x+1;
P_3(x) = (x-1)*(x^2-1)*(x^3-1) = x^6-x^5-x^4+x^2+x-1;
P_4(x) = (x-1)*(x^2-1)*(x^3-1)*(x^4-1) = x^10 - x^9 - x^8+2x^5-x^2-x+1, hence a(2)=4.
n=3 first appears as a coefficient of P_{10}(x).

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Dorin Andrica and Ovidiu Bagdasar, On some results concerning the polygonal polynomials, Carpathian Journal of Mathematics (2019) Vol. 35, No. 1, 1-11.
Jiro Suzuki, On coefficients of cyclotomic polynomials, Proc. Japan Acad. Ser. A Math. Sci. 63:7 (1987), pp. 279-280.

Cf. A231599: a(n) is the index of the first row m containing number n.

T:= proc(n) option remember; [(p-> seq(coeff(p, x, i),
    i=0..degree(p)))(expand(mul(1-x^i, i=1..n)))] end:
a:= proc(n) local k; for k while not n in T(k) do od: k end:
seq(a(n), n=0..70);  # Alois P. Heinz, Mar 29 2019

With[{s = Array[CoefficientList[Times @@ Array[x^# - 1 &, #], x] &, 40]}, TakeWhile[Array[FirstPosition[s, #][[1]] &, Max@ Map[Max, s]], IntegerQ]] (* Michael De Vlieger, Apr 05 2018 *)

a(n) = {my(k=1); while (!vecsearch(vecsort(Vec(prod(j=1, k, x^j-1))), n), k++); k;} \\ Michel Marcus, Apr 08 2018

first(n) = {my(pol = [1], res = vector(n), todo = n+1, t = 0); while(1, t++; for(i = 1, #pol, if(0 < pol[i] && pol[i] <=n, if(res[pol[i]] == 0, res[pol[i]] = t-1; todo--; if(todo == 0, return(concat([3], res))))));  pol = concat(pol, vector(t)) - concat(vector(t), pol))} \\ David A. Corneth, Apr 08 2018

Offset changed to 0 by David A. Corneth, Apr 08 2018

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