Real-rooted atlas

A074246 → D+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel

Recurrence

\(P_{n}(x) = \left(-1 + 3\,x - 3\,x^{2} + x^{3} + 3\,n - 6\,n x + 3\,n x^{2} - 3\,n^{2} + 3\,n^{2} x + n^{3}\right)\,P_{n-3}(x) + \left(-1 + 3\,x - 3\,x^{2} + 3\,n - 6\,n x - 3\,n^{2}\right)\,P_{n-2}(x) + \left(3\,x + 3\,n\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 5 + 2\,x\)
\(P_{2}(x) = 23 + 18\,x + 3\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1527
22318344
3120130424296
47249604358052204
55012765843051120135618236
63933266878438271428024502107166984
734583263995247124018051639060478830881681704
83371976668748054029402342760599865932408610432918507312
936135792759641526623166031708820924346517089382009701452058510221208912
1042237979293338440886772298845077362014582441530837114433148139996023265770112855677824
1153495619841234720843212128526696675023034423800576805608600149045636610011144746460357509901239617695872
12729961931521750759526881804337043601065531907444040888852010417190784187875587923951928021468447132132053053124813588026239488
131067779243008265024915804828496270493601772153757920715596521640199076996118394012268655633776720583287705432632202237235764401547149300146595840
1416670798231040426736094523844765274725867231020695524000132324568558003931326705306842045074871132483917280154231248601324543220827246423647280107380189015156172997168640
Technical details
Node id
n-5f739b5cc53b5208 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-b4083b2ff491c73b

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
  • Palindromic rows: no
  • γ-positive (gamma_positive): no
  • Nonnegative coefficients (nonn): yes
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:        1;
  1:        5,        2;
  2:       23,       18,        3;
  3:      120,      130,       42,        4;
  4:      724,      960,      435,       80,       5;
  5:     5012,     7658,     4305,     1120,     135,       6;
  6:    39332,    66878,    43827,    14280,    2450,     210,      7;
  7:   345832,   639952,   471240,   180516,   39060,    4788,    308,     8;
  8:  3371976,  6687480,  5402940,  2342760,  599865,   93240,   8610,   432,   9;
  9: 36135792, 75964152, 66231660, 31708820, 9243465, 1708938, 200970, 14520, 585, 10;
%F Formula
The row polynomials satisfy P(n,x) = (n^3-3*n^2+3*n-1+(3*n^2-6*n+3)*x+(3*n-3)*x^2+x^3)*P(n-3,x) - (3*n^2-3*n+1+(6*n-3)*x+3*x^2)*P(n-2,x) + (3*n+3*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 5 + 2*x, P(2,x) = 23 + 18*x + 3*x^2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 5 + 2*x;
P[2, x_] := 23 + 18*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[(n^3 - 3*n^2 + 3*n - 1 + (3*n^2 - 6*n + 3)*x + (3*n - 3)*x^2 + x^3)*P[n - 3, x] - (3*n^2 - 3*n + 1 + (6*n - 3)*x + 3*x^2)*P[n - 2, x] + (3*n + 3*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 5 + 2*x; v[3] = 23 + 18*x + 3*x^2; for(n = 3, N, my(i = n + 1); v[i] = (n^3 - 3*n^2 + 3*n - 1 + (3*n^2 - 6*n + 3)*x + (3*n - 3)*x^2 + x^3)*v[i - 3] - (3*n^2 - 3*n + 1 + (6*n - 3)*x + 3*x^2)*v[i - 2] + (3*n + 3*x)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Proof chains ? JSON

Same fingerprint ?

References

  1. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.