Real-rooted atlas

A094305 → x ↦ x+1 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-x + x^{3} - n + 3\,n x^{2} + 3\,n^{2} x + n^{3}\right)\,P_{n}(x) = \left(16\,x + 48\,x^{2} + 56\,x^{3} + 32\,x^{4} + 9\,x^{5} + x^{6} + 16\,n + 72\,n x + 108\,n x^{2} + 74\,n x^{3} + 24\,n x^{4} + 3\,n x^{5} + 24\,n^{2} + 60\,n^{2} x + 54\,n^{2} x^{2} + 21\,n^{2} x^{3} + 3\,n^{2} x^{4} + 8\,n^{3} + 12\,n^{3} x + 6\,n^{3} x^{2} + n^{3} x^{3}\right)\,P_{n-3}(x) + \left(24 + 36\,x - 6\,x^{2} - 33\,x^{3} - 18\,x^{4} - 3\,x^{5} + 12\,n - 36\,n x - 81\,n x^{2} - 48\,n x^{3} - 9\,n x^{4} - 24\,n^{2} - 60\,n^{2} x - 42\,n^{2} x^{2} - 9\,n^{2} x^{3} - 12\,n^{3} - 12\,n^{3} x - 3\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-12\,x + 9\,x^{3} + 3\,x^{4} - 12\,n + 6\,n x + 24\,n x^{2} + 9\,n x^{3} + 6\,n^{2} + 21\,n^{2} x + 9\,n^{2} x^{2} + 6\,n^{3} + 3\,n^{3} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 9 + 3\,x\)
\(P_{2}(x) = 48 + 36\,x + 6\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
19312
24836690
32002409010540
47201200720180152835
523525040420016803152113608
6716818816201601120033605042861236
72073664512846726048025200604875636262440
85760020736032256028224015120050400100801080451082565
91548806336001140480118272077616033264092400158401485554330260
10405504185856038016004561920354816018627846652801584002376019806616888014
11103833652715521208064016473600148262409225216403603212355202574003432025747864481508
1226091521453670436900864563763205765760041513472215255048072064216216040040048048327691241805655
1364512003913728010902528018450432021141120017297280010378368046126080151351203603600600600655204095105892820880
1415728640103219200313098240581468160738017280676515840461260800237219840922521602690688057657608736008736050401203252418920
Technical details
Node id
n-c11a77a101be52d0 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-49d880ced4b36b3a

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:      1;
  1:      9,      3;
  2:     48,     36,       6;
  3:    200,    240,      90,      10;
  4:    720,   1200,     720,     180,     15;
  5:   2352,   5040,    4200,    1680,    315,     21;
  6:   7168,  18816,   20160,   11200,   3360,    504,    28;
  7:  20736,  64512,   84672,   60480,  25200,   6048,   756,    36;
  8:  57600, 207360,  322560,  282240, 151200,  50400, 10080,  1080,   45;
  9: 154880, 633600, 1140480, 1182720, 776160, 332640, 92400, 15840, 1485, 55;
%F Formula
The row polynomials satisfy (n^3-n+(3*n^2-1)*x+3*n*x^2+x^3)*P(n,x) = (8*n^3+24*n^2+16*n+(12*n^3+60*n^2+72*n+16)*x+(6*n^3+54*n^2+108*n+48)*x^2+(n^3+21*n^2+74*n+56)*x^3+(3*n^2+24*n+32)*x^4+(3*n+9)*x^5+x^6)*P(n-3,x) - (12*n^3+24*n^2-12*n-24+(12*n^3+60*n^2+36*n-36)*x+(3*n^3+42*n^2+81*n+6)*x^2+(9*n^2+48*n+33)*x^3+(9*n+18)*x^4+3*x^5)*P(n-2,x) + (6*n^3+6*n^2-12*n+(3*n^3+21*n^2+6*n-12)*x+(9*n^2+24*n)*x^2+(9*n+9)*x^3+3*x^4)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 9 + 3*x, P(2,x) = 48 + 36*x + 6*x^2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 9 + 3*x;
P[2, x_] := 48 + 36*x + 6*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((8*n^3 + 24*n^2 + 16*n + (12*n^3 + 60*n^2 + 72*n + 16)*x + (6*n^3 + 54*n^2 + 108*n + 48)*x^2 + (n^3 + 21*n^2 + 74*n + 56)*x^3 + (3*n^2 + 24*n + 32)*x^4 + (3*n + 9)*x^5 + x^6)*P[n - 3, x] - (12*n^3 + 24*n^2 - 12*n - 24 + (12*n^3 + 60*n^2 + 36*n - 36)*x + (3*n^3 + 42*n^2 + 81*n + 6)*x^2 + (9*n^2 + 48*n + 33)*x^3 + (9*n + 18)*x^4 + 3*x^5)*P[n - 2, x] + (6*n^3 + 6*n^2 - 12*n + (3*n^3 + 21*n^2 + 6*n - 12)*x + (9*n^2 + 24*n)*x^2 + (9*n + 9)*x^3 + 3*x^4)*P[n - 1, x])/(n^3 - n + (3*n^2 - 1)*x + 3*n*x^2 + x^3)]];
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] = 9 + 3*x; v[3] = 48 + 36*x + 6*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((8*n^3 + 24*n^2 + 16*n + (12*n^3 + 60*n^2 + 72*n + 16)*x + (6*n^3 + 54*n^2 + 108*n + 48)*x^2 + (n^3 + 21*n^2 + 74*n + 56)*x^3 + (3*n^2 + 24*n + 32)*x^4 + (3*n + 9)*x^5 + x^6)*v[i - 3] - (12*n^3 + 24*n^2 - 12*n - 24 + (12*n^3 + 60*n^2 + 36*n - 36)*x + (3*n^3 + 42*n^2 + 81*n + 6)*x^2 + (9*n^2 + 48*n + 33)*x^3 + (9*n + 18)*x^4 + 3*x^5)*v[i - 2] + (6*n^3 + 6*n^2 - 12*n + (3*n^3 + 21*n^2 + 6*n - 12)*x + (9*n^2 + 24*n)*x^2 + (9*n + 9)*x^3 + 3*x^4)*v[i - 1])/(n^3 - n + (3*n^2 - 1)*x + 3*n*x^2 + x^3)); 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

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  2. [AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
  3. [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
  4. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  5. [Edr53] Albert Edrei. Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence. Canadian Journal of Mathematics, 5:86–94, 1953. doi:10.4153/CJM-1953-010-3
  6. [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
  7. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  8. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.