Real-rooted atlas

A013613 → x ↦ x+1 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(49 - 42\,x + 126\,n x + 36\,n^{2} x^{2}\right)\,P_{n}(x) = \left(343 + 882\,x + 756\,x^{2} + 216\,x^{3} + 882\,n x + 1260\,n x^{2} + 432\,n x^{3} + 252\,n^{2} x^{2} + 216\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
172431
249336324709
334335286804345614131
4240132928952569676832400259753
5168072881201111320169344011340002799364523623
6117649242020811668860237081602381400011757312228614475772333
782354319765032114354828290424960388962000288054144112021056179159041232321467
8576480115812025610673117283252759552544546800053770106883136589568100329062413604889619582364113
94035360712451970169605805552341539752966861289680084687918336658683809283160365465685710804481007769600305397032239
10282475249968486568084050798580341539752960800483796000118563085670411526966662407374186086402999878156807054387200073164072964689635915029
1119773267437457346573671910127674032873201222408805321756000152155959943681775152866009614195308216320769968726912027159390720005633633617925224277606471081959297219
121384128720156947010198460404507246163068165447424092455878438000182587151932416248521401241344238481178034176161693432651520760462940160002366126119526443883931893763678762147841065508283500921
13968890104074318481606712499710014490962792030557155849348316597620002076928853231232323077821613747236169645335183362942820474257664173005318886400071772492292300819967189011660833476735545344255989602713615819399798164599
1467822307284932554707496752408096511834284248744540546611291613502656676002261544751296230439577033147684032506375034692567044806606774620851233909042501734400175842606116136966522615077142528164036004172185625086981065932817631936921600232910956969442557
Technical details
Node id
n-d665c3dd5d9a2622 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-950975dd37953e66

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,         24;
  2:       49,        336,        324;
  3:      343,       3528,       6804,        3456;
  4:     2401,      32928,      95256,       96768,       32400;
  5:    16807,     288120,    1111320,     1693440,     1134000,      279936;
  6:   117649,    2420208,   11668860,    23708160,    23814000,    11757312,     2286144;
  7:   823543,   19765032,  114354828,   290424960,   388962000,   288054144,   112021056,    17915904;
  8:  5764801,  158120256, 1067311728,  3252759552,  5445468000,  5377010688,  3136589568,  1003290624,  136048896;
  9: 40353607, 1245197016, 9605805552, 34153975296, 68612896800, 84687918336, 65868380928, 31603654656, 8571080448, 1007769600;
%F Formula
The row polynomials satisfy (49+(126*n-42)*x+36*n^2*x^2)*P(n,x) = (343+(882*n+882)*x+(252*n^2+1260*n+756)*x^2+(216*n^2+432*n+216)*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 + (882*n + 882)*x + (252*n^2 + 1260*n + 756)*x^2 + (216*n^2 + 432*n + 216)*x^3)*P[n - 1, x])/(49 + (126*n - 42)*x + 36*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 + (882*n + 882)*x + (252*n^2 + 1260*n + 756)*x^2 + (216*n^2 + 432*n + 216)*x^3)*v[i - 1])/(49 + (126*n - 42)*x + 36*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.