Real-rooted atlas

A034870 → Veronese 2 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-9 + 3\,x + 9\,n - 26\,n x + 45\,n^{2} x - n^{2} x^{2} - 22\,n^{3} x + 5\,n^{3} x^{2} - 8\,n^{4} x^{2} + 4\,n^{5} x^{2}\right)\,P_{n}(x) = \left(-9 - 105\,x - 75\,x^{2} - 3\,x^{3} + 9\,n + 82\,n x + 98\,n x^{2} + 11\,n x^{3} + 45\,n^{2} x + 90\,n^{2} x^{2} - 3\,n^{2} x^{3} - 22\,n^{3} x - 109\,n^{3} x^{2} - 25\,n^{3} x^{3} - 8\,n^{4} x^{2} + 8\,n^{4} x^{3} + 4\,n^{5} x^{2} + 12\,n^{5} x^{3}\right)\,P_{n-1}(x) + \left(18\,x + 10\,x^{2} - 26\,x^{3} - 2\,x^{4} - 20\,n x^{2} + 14\,n x^{3} + 6\,n x^{4} - 44\,n^{2} x^{2} + 42\,n^{2} x^{3} + 2\,n^{2} x^{4} + 16\,n^{3} x^{3} - 16\,n^{3} x^{4} + 8\,n^{4} x^{3} - 8\,n^{4} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1145
2124934
316013516212
41112630448251216
51180189033601125366592
612644455147841237523764934304
713649009480487507536036445964173056
814801638012812832175028828889180768081851968
916122754029702410939501575288909636195840123931004112384
1017604360562016031492506651216617253024806403924451900012119529728
111924658351193808799425023279256316856542046528060436537315002795114491488256
121110495634215353618386775706052161325036441255203845957315113459600128574639744169423624704
1311300134550368368039056875191222460473227300618092800430250535156227500278578302152800549251961941962752
14115121842756027840777026254724319601490665995256746240024641621551312311000376080705542505603460275740882258824815616
Technical details
Node id
n-35489c3e82399690 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-395f2289776a8710

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: 1,   4;
  2: 1,  24,     9;
  3: 1,  60,   135,     16;
  4: 1, 112,   630,    448,      25;
  5: 1, 180,  1890,   3360,    1125,      36;
  6: 1, 264,  4455,  14784,   12375,    2376,     49;
  7: 1, 364,  9009,  48048,   75075,   36036,   4459,     64;
  8: 1, 480, 16380, 128128,  321750,  288288,  89180,   7680,    81;
  9: 1, 612, 27540, 297024, 1093950, 1575288, 909636, 195840, 12393, 100;
%F Formula
The row polynomials satisfy (9*n-9-(22*n^3-45*n^2+26*n-3)*x+(4*n^5-8*n^4+5*n^3-n^2)*x^2)*P(n,x) = (9*n-9-(22*n^3-45*n^2-82*n+105)*x+(4*n^5-8*n^4-109*n^3+90*n^2+98*n-75)*x^2+(12*n^5+8*n^4-25*n^3-3*n^2+11*n-3)*x^3)*P(n-1,x) + (18*x-(44*n^2+20*n-10)*x^2+(8*n^4+16*n^3+42*n^2+14*n-26)*x^3-(8*n^4+16*n^3-2*n^2-6*n+2)*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 4*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[((9*n - 9 - (22*n^3 - 45*n^2 - 82*n + 105)*x + (4*n^5 - 8*n^4 - 109*n^3 + 90*n^2 + 98*n - 75)*x^2 + (12*n^5 + 8*n^4 - 25*n^3 - 3*n^2 + 11*n - 3)*x^3)*P[n - 1, x] + (18*x - (44*n^2 + 20*n - 10)*x^2 + (8*n^4 + 16*n^3 + 42*n^2 + 14*n - 26)*x^3 - (8*n^4 + 16*n^3 - 2*n^2 - 6*n + 2)*x^4)*D[P[n - 1, x], x])/(9*n - 9 - (22*n^3 - 45*n^2 + 26*n - 3)*x + (4*n^5 - 8*n^4 + 5*n^3 - n^2)*x^2)]];
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] = 1 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = ((9*n - 9 - (22*n^3 - 45*n^2 - 82*n + 105)*x + (4*n^5 - 8*n^4 - 109*n^3 + 90*n^2 + 98*n - 75)*x^2 + (12*n^5 + 8*n^4 - 25*n^3 - 3*n^2 + 11*n - 3)*x^3)*v[i - 1] + (18*x - (44*n^2 + 20*n - 10)*x^2 + (8*n^4 + 16*n^3 + 42*n^2 + 14*n - 26)*x^3 - (8*n^4 + 16*n^3 - 2*n^2 - 6*n + 2)*x^4)*deriv(v[i - 1], x))/(9*n - 9 - (22*n^3 - 45*n^2 + 26*n - 3)*x + (4*n^5 - 8*n^4 + 5*n^3 - n^2)*x^2)); 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. [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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.