Real-rooted atlas

A008278 → x ↦ x+1 (shift +1) → x ↦ x+1 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-4\,x + n x^{3}\right)\,P_{n}(x) = \left(-32 - 60\,x - 16\,x^{2} + 15\,x^{3} + 6\,x^{4} + 64\,n + 64\,n x - 5\,n x^{3} + 4\,n x^{4} + n x^{5} + 16\,n^{2} x - 16\,n^{2} x^{3} - 6\,n^{2} x^{4} - 4\,n^{3} x^{3} - 4\,n^{3} x^{4} - n^{3} x^{5}\right)\,P_{n-2}(x) + \left(-64 - 112\,x - 48\,x^{2} + 16\,x^{3} + 16\,x^{4} + 3\,x^{5} - 32\,n x - 32\,n x^{2} + 12\,n x^{3} + 24\,n x^{4} + 9\,n x^{5} + n x^{6} + 8\,n^{2} x^{3} + 12\,n^{2} x^{4} + 6\,n^{2} x^{5} + n^{2} x^{6}\right)\,P_{n-2}'(x) + \left(-32 - 16\,x + 8\,x^{2} + 3\,x^{3} - 16\,n x + 6\,n x^{3} + 4\,n^{2} x^{3} + 2\,n^{2} x^{4}\right)\,P_{n-1}(x) + \left(16\,x + 16\,x^{2} + 4\,x^{3} - 4\,n x^{3} - 4\,n x^{4} - n x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1325
21114328
34992394184
42576444179251415
51539485442871512205612403
61029939290442322175249554507120985
775905340472467466296712963051544498781293299
86094413147464511174239905841693355395628469492168914997340
952844513091927458083474539687362841564587986941558403141200476110187174648
104913492332153815068677673774155542446765976518133276243191106600030442385510430112497623467
114870269293527383892845072448910412385516766695989535838375601079130780204748480228487411276860227591235436345913
12512090544140693177420108152680887149900251572126411350110693227410322533079976061355887369503987498667690038717694942813532108491817
1356878092067492272324062143827586626522166778118482108898978660132941103049257268272488016947724492833854898663435910935032905176711830392106821148423129070209
1466492002181962285900593461985300320414233702029734808357381561272902549058524938812670036943830444244980249610918334648791832200401701987837284412536733843641849922975415140085993342664
Technical details
Node id
n-91fef0adbb55a62f · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-5545d9de5dd74d0a

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:       3,        2;
  2:      11,       14,        3;
  3:      49,       92,       39,        4;
  4:     257,      644,      417,       92,        5;
  5:    1539,     4854,     4287,     1512,      205,       6;
  6:   10299,    39290,    44232,    21752,     4955,     450,       7;
  7:   75905,   340472,   467466,   296712,    96305,   15444,     987,      8;
  8:  609441,  3147464,  5111742,  3990584,  1693355,  395628,   46949,   2168,    9;
  9: 5284451, 30919274, 58083474, 53968736, 28415645, 8798694, 1558403, 141200, 4761, 10;
%F Formula
The row polynomials satisfy -(4*x-n*x^3)*P(n,x) = (64*n-32+(16*n^2+64*n-60)*x-16*x^2-(4*n^3+16*n^2+5*n-15)*x^3-(4*n^3+6*n^2-4*n-6)*x^4-(n^3-n)*x^5)*P(n-2,x) - (64+(32*n+112)*x+(32*n+48)*x^2-(8*n^2+12*n+16)*x^3-(12*n^2+24*n+16)*x^4-(6*n^2+9*n+3)*x^5-(n^2+n)*x^6)*d/dx P(n-2,x) - (32+(16*n+16)*x-8*x^2-(4*n^2+6*n+3)*x^3-2*n^2*x^4)*P(n-1,x) + (16*x+16*x^2-(4*n-4)*x^3-4*n*x^4-n*x^5)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((64*n - 32 + (16*n^2 + 64*n - 60)*x - 16*x^2 - (4*n^3 + 16*n^2 + 5*n - 15)*x^3 - (4*n^3 + 6*n^2 - 4*n - 6)*x^4 - (n^3 - n)*x^5)*P[n - 2, x] - (64 + (32*n + 112)*x + (32*n + 48)*x^2 - (8*n^2 + 12*n + 16)*x^3 - (12*n^2 + 24*n + 16)*x^4 - (6*n^2 + 9*n + 3)*x^5 - (n^2 + n)*x^6)*D[P[n - 2, x], x] - (32 + (16*n + 16)*x - 8*x^2 - (4*n^2 + 6*n + 3)*x^3 - 2*n^2*x^4)*P[n - 1, x] + (16*x + 16*x^2 - (4*n - 4)*x^3 - 4*n*x^4 - n*x^5)*D[P[n - 1, x], x])/(-4*x + n*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] = 3 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = ((64*n - 32 + (16*n^2 + 64*n - 60)*x - 16*x^2 - (4*n^3 + 16*n^2 + 5*n - 15)*x^3 - (4*n^3 + 6*n^2 - 4*n - 6)*x^4 - (n^3 - n)*x^5)*v[i - 2] - (64 + (32*n + 112)*x + (32*n + 48)*x^2 - (8*n^2 + 12*n + 16)*x^3 - (12*n^2 + 24*n + 16)*x^4 - (6*n^2 + 9*n + 3)*x^5 - (n^2 + n)*x^6)*deriv(v[i - 2], x) - (32 + (16*n + 16)*x - 8*x^2 - (4*n^2 + 6*n + 3)*x^3 - 2*n^2*x^4)*v[i - 1] + (16*x + 16*x^2 - (4*n - 4)*x^3 - 4*n*x^4 - n*x^5)*deriv(v[i - 1], x))/(-4*x + n*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

References

  1. [ASW52] Michael Aissen, I. J. Schoenberg and A. M. Whitney. On the generating functions of totally positive sequences I. Journal d’Analyse Mathématique, 2(1):93–103, 1952. doi:10.1007/BF02786970
  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.