Real-rooted atlas

A008278 → x ↦ x+1 (shift +1) → reverse → D+1

nonnegative coefficients

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

Recurrence

\(\left(6\right)\,P_{n}(x) = \left(-21 - 97\,x - 167\,x^{2} - 90\,x^{3} - 2\,x^{4} + 8\,x^{5} + 2\,n + 21\,n x + 72\,n x^{2} + 72\,n x^{3} + 17\,n x^{4} - 4\,n x^{5} - 6\,n^{2} x^{2} - 12\,n^{2} x^{3} - 6\,n^{2} x^{4}\right)\,P_{n-2}(x) + \left(-3 - 26\,x - 76\,x^{2} - 92\,x^{3} - 47\,x^{4} - 3\,x^{5} + 4\,x^{6} + n x + 10\,n x^{2} + 24\,n x^{3} + 23\,n x^{4} + 8\,n x^{5}\right)\,P_{n-2}'(x) + \left(-x - 5\,x^{2} - 10\,x^{3} - 11\,x^{4} - 7\,x^{5} - 2\,x^{6}\right)\,P_{n-2}''(x) + \left(17 + 55\,x + 19\,x^{2} - 4\,x^{3} - 12\,n x - 12\,n x^{2}\right)\,P_{n-1}(x) + \left(1 + 9\,x + 15\,x^{2} + 8\,x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1325
2615526
311566815150
42017140331752963
5374841817270715592036807
670133172431676818021813887752448
713536322716288827146129121886450934140437004
82649939988314320159883261249015847029264671211473911237
9521274283548752001115601919210468201106769456079691164093011597537384873
10103476539126987190153063438026577772623108352392922781604517537510714943678570379715078
11205921616045545964001177118863229453499607296217067711131067578114741223477712067348189142135974080631202
124108617579164294111763764891009288276349792286278261263021162191102211459587703728790706427735107305278560402764443746225182023
1382051782884597212617763655875319069045221062561675995493619011012810079013878801065511897263191266994595678229025267403962222471190899322550157126907
141639851927392189943713424637700277990145251365820989444405364350479751600251321513427011660165243383220112507209110016311382479461956673072003100861420313829585456859505297612
Technical details
Node id
n-4713d082046abed1 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-34ec7846500b9da6

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:   6,    15,      5;
  3:  11,    56,     68,      15;
  4:  20,   171,    403,     317,      52;
  5:  37,   484,   1817,    2707,    1559,      203;
  6:  70,  1331,   7243,   16768,   18021,     8138,      877;
  7: 135,  3632,  27162,   88827,  146129,   121886,    45093,    4140;
  8: 264,  9939,  98831,  432015,  988326,  1249015,   847029,  264671,   21147;
  9: 521, 27428, 354875, 2001115, 6019192, 10468201, 10676945, 6079691, 1640930, 115975;
%F Formula
The row polynomials satisfy 6*P(n,x) = (2*n-21+(21*n-97)*x-(6*n^2-72*n+167)*x^2-(12*n^2-72*n+90)*x^3-(6*n^2-17*n+2)*x^4-(4*n-8)*x^5)*P(n-2,x) - (3-(n-26)*x-(10*n-76)*x^2-(24*n-92)*x^3-(23*n-47)*x^4-(8*n-3)*x^5-4*x^6)*d/dx P(n-2,x) - (x+5*x^2+10*x^3+11*x^4+7*x^5+2*x^6)*d^2/dx^2 P(n-2,x) + (17-(12*n-55)*x-(12*n-19)*x^2-4*x^3)*P(n-1,x) + (1+9*x+15*x^2+8*x^3)*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[((2*n - 21 + (21*n - 97)*x - (6*n^2 - 72*n + 167)*x^2 - (12*n^2 - 72*n + 90)*x^3 - (6*n^2 - 17*n + 2)*x^4 - (4*n - 8)*x^5)*P[n - 2, x] - (3 - (n - 26)*x - (10*n - 76)*x^2 - (24*n - 92)*x^3 - (23*n - 47)*x^4 - (8*n - 3)*x^5 - 4*x^6)*D[P[n - 2, x], x] - (x + 5*x^2 + 10*x^3 + 11*x^4 + 7*x^5 + 2*x^6)*D[P[n - 2, x], {x, 2}] + (17 - (12*n - 55)*x - (12*n - 19)*x^2 - 4*x^3)*P[n - 1, x] + (1 + 9*x + 15*x^2 + 8*x^3)*D[P[n - 1, x], x])/(6)]];
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] = ((2*n - 21 + (21*n - 97)*x - (6*n^2 - 72*n + 167)*x^2 - (12*n^2 - 72*n + 90)*x^3 - (6*n^2 - 17*n + 2)*x^4 - (4*n - 8)*x^5)*v[i - 2] - (3 - (n - 26)*x - (10*n - 76)*x^2 - (24*n - 92)*x^3 - (23*n - 47)*x^4 - (8*n - 3)*x^5 - 4*x^6)*deriv(v[i - 2], x) - (x + 5*x^2 + 10*x^3 + 11*x^4 + 7*x^5 + 2*x^6)*deriv(deriv(v[i - 2], x), x) + (17 - (12*n - 55)*x - (12*n - 19)*x^2 - 4*x^3)*v[i - 1] + (1 + 9*x + 15*x^2 + 8*x^3)*deriv(v[i - 1], x))/(6)); 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.