Real-rooted atlas

A131431 → θ+1 → basis A013611

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(10 - 21\,n + 9\,n^{2}\right)\,P_{n}(x) = \left(2 + 16\,x + 32\,x^{2} + 3\,n + 24\,n x + 48\,n x^{2} - 9\,n^{2} - 72\,n^{2} x - 144\,n^{2} x^{2}\right)\,P_{n-2}(x) + \left(-10 - 40\,x - 24\,n - 96\,n x + 18\,n^{2} + 72\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 8 + 32\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
183240
221168336525
340480192025605000
46510406240166401664040625
5961920153606144012288098304300000
61333192319201702405107208171525447682078125
7176492859136394240157696037847045046272288358413750000
8225720010080080640040320001290240025804800294912001474560087890625
92801008016128015052809031680361267209633792016515072016515072073400320546875000
10341136402455202618880183321608799436829331456067043328010056499208939110403575644163330078125
11408179523590404308480344678401930199047720796162205941760441188352058825113604706009088171127603219921875000
124812308850793667724806095232039009484818204426246241517568156037939202774007808033288093696242095226888069840896117431640625
1356029120698880102502401025024007380172803936092160157443686404723310592010496245760016793993216018320719872012213813248037580963840683593750000
14645361209391201502592016528512013222809607933685760362682777601269389721603385039257606770078515209847386931209847386931206059930419201731408691203936767578125
Technical details
Node id
n-30d2ffb507fad587 · revision 0 · JSON
Definition source
derived:basis_map_A013611
Fingerprint
f6-03d7b8417365c509

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:   8,    32;
  2:  21,   168,    336;
  3:  40,   480,   1920,    2560;
  4:  65,  1040,   6240,   16640,   16640;
  5:  96,  1920,  15360,   61440,  122880,    98304;
  6: 133,  3192,  31920,  170240,  510720,   817152,   544768;
  7: 176,  4928,  59136,  394240, 1576960,  3784704,  5046272,   2883584;
  8: 225,  7200, 100800,  806400, 4032000, 12902400, 25804800,  29491200,  14745600;
  9: 280, 10080, 161280, 1505280, 9031680, 36126720, 96337920, 165150720, 165150720, 73400320;
%F Formula
The row polynomials satisfy (9*n^2-21*n+10)*P(n,x) = -(9*n^2-3*n-2+(72*n^2-24*n-16)*x+(144*n^2-48*n-32)*x^2)*P(n-2,x) + (18*n^2-24*n-10+(72*n^2-96*n-40)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 8 + 32*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 8 + 32*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(9*n^2 - 3*n - 2 + (72*n^2 - 24*n - 16)*x + (144*n^2 - 48*n - 32)*x^2)*P[n - 2, x] + (18*n^2 - 24*n - 10 + (72*n^2 - 96*n - 40)*x)*P[n - 1, x])/(9*n^2 - 21*n + 10)]];
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] = 8 + 32*x; for(n = 2, N, my(i = n + 1); v[i] = (-(9*n^2 - 3*n - 2 + (72*n^2 - 24*n - 16)*x + (144*n^2 - 48*n - 32)*x^2)*v[i - 2] + (18*n^2 - 24*n - 10 + (72*n^2 - 96*n - 40)*x)*v[i - 1])/(9*n^2 - 21*n + 10)); 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. [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.