Real-rooted atlas

A038255 → basis A013609 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(49 - 14\,x + 42\,n x + 4\,n^{2} x^{2}\right)\,P_{n}(x) = \left(343 + 294\,x + 84\,x^{2} + 8\,x^{3} + 294\,n x + 140\,n x^{2} + 16\,n x^{3} + 28\,n^{2} x^{2} + 8\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
17815
24911236197
334311767561282403
424011097610584358440027945
5168079604012348062720140001152314199
611764980673612965408780802940004838431363444525
78235436588344127060921075648048020001185408153664819237023723
85764801527067521185901921204725766722800022127616430259245875220736391672017
9403536074150656721067311728126496204884707280034850995290354432144506881306368512004089438495
10282475249322828856093389776201264962048098825160004879139328158120256033718272045722880358400012390442228833301
11197732674324857821912799001418601217525971201087076760006261562137624350519424649076736011735539201379840009540608294912431973845235
121384128720118982336732867116119162411363575731201141430598000751387456512340907271936109044891648246446323203863552000400705536247726086922244382887992057
13968890104071439493868904555233349434410340853915392115411316020008547032317824443179453516816538475233284485323082248789580800012154734592112715366462992384160563244153150869863
14678223072849108515691655844534405687047692127607609856113103089699600930676852385285428948305580823153865326592732602770099217227578368002977909975043682035302430866268161573519363686400442002224592765
Technical details
Node id
n-7a1b67fb7e683084 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-2b0c83ee9d9e3a42

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:        7,         8;
  2:       49,       112,         36;
  3:      343,      1176,        756,        128;
  4:     2401,     10976,      10584,       3584,       400;
  5:    16807,     96040,     123480,      62720,     14000,      1152;
  6:   117649,    806736,    1296540,     878080,    294000,     48384,     3136;
  7:   823543,   6588344,   12706092,   10756480,   4802000,   1185408,   153664,     8192;
  8:  5764801,  52706752,  118590192,  120472576,  67228000,  22127616,  4302592,   458752,   20736;
  9: 40353607, 415065672, 1067311728, 1264962048, 847072800, 348509952, 90354432, 14450688, 1306368, 51200;
%F Formula
The row polynomials satisfy (49+(42*n-14)*x+4*n^2*x^2)*P(n,x) = (343+(294*n+294)*x+(28*n^2+140*n+84)*x^2+(8*n^2+16*n+8)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((343 + (294*n + 294)*x + (28*n^2 + 140*n + 84)*x^2 + (8*n^2 + 16*n + 8)*x^3)*P[n - 1, x])/(49 + (42*n - 14)*x + 4*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; for(n = 1, N, my(i = n + 1); v[i] = ((343 + (294*n + 294)*x + (28*n^2 + 140*n + 84)*x^2 + (8*n^2 + 16*n + 8)*x^3)*v[i - 1])/(49 + (42*n - 14)*x + 4*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. [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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.