Real-rooted atlas

A038326 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(121 - 132\,x + 396\,n x + 144\,n^{2} x^{2}\right)\,P_{n}(x) = \left(1331 + 4356\,x + 4752\,x^{2} + 1728\,x^{3} + 4356\,n x + 7920\,n x^{2} + 3456\,n x^{3} + 1584\,n^{2} x^{2} + 1728\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1114859
2121105612962473
3133117424427682764889171
41464125555294089612165125184002946001
51610513513840172497603345408028512000895795291848683
61771561463826882846210407359897609408960005912248321463132162747199097
719487171595244496438316401614167802880241496640002276215603211266117632229323571279636871939
8214358881748307366464286405568249353330688531292608000667689910272495709175808201804742656348285173762252662122913
923579476919260303659290919345017641143299563521051959363840016525325279232163584028016649989334761472344802322022451597803520062475142127003
1025937424601113181489168012501409939920646537564569601928592167040003635571561431044498560770457603662756079206401896412771123205675758387200074920010711041704751838582089
11285311670611136949601893281680745114144809778880664115203333709317312000733173598221926410886517064507392110798371395993607648864843530240343383382425600090653212960358410699320537907245887966356092851
12313842837672116433952227193622185835506711361434235830736896055006203735648000138255592807563264239503375419162624292507700485423104252412539836497920151088688267264000598311205538365441412310311003750415068209757552641220963564999876977
133452271214393119583793070739042884158615872476820509572379537612887398745935529600024713187214351933444892711812134322176697143352823591731272189986393238405125401420605554688000285195007973287526410098018723676815362154753995330027522097066825429811232163998997900433739
1437974983358324123199262560721632370133689036967856287134013313526579213459406874071558400422870092334466416649418470238358570188815337153762119018086418528763174264523980816636375465108439040010980007806971569766451836496114874318848165916057640412119043229482911161909248288881654523494400839965639854906883609
Technical details
Node id
n-0cf1f47b9fa11bc5 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-1bffe7ba5171bd1a

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:         11,          48;
  2:        121,        1056,         1296;
  3:       1331,       17424,        42768,         27648;
  4:      14641,      255552,       940896,       1216512,         518400;
  5:     161051,     3513840,     17249760,      33454080,       28512000,        8957952;
  6:    1771561,    46382688,    284621040,     735989760,      940896000,      591224832,      146313216;
  7:   19487171,   595244496,   4383164016,   14167802880,    24149664000,    22762156032,    11266117632,    2293235712;
  8:  214358881,  7483073664,  64286405568,  249353330688,   531292608000,   667689910272,   495709175808,  201804742656,   34828517376;
  9: 2357947691, 92603036592, 909193450176, 4114329956352, 10519593638400, 16525325279232, 16358402801664, 9989334761472, 3448023220224, 515978035200;
%F Formula
The row polynomials satisfy (121+(396*n-132)*x+144*n^2*x^2)*P(n,x) = (1331+(4356*n+4356)*x+(1584*n^2+7920*n+4752)*x^2+(1728*n^2+3456*n+1728)*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[((1331 + (4356*n + 4356)*x + (1584*n^2 + 7920*n + 4752)*x^2 + (1728*n^2 + 3456*n + 1728)*x^3)*P[n - 1, x])/(121 + (396*n - 132)*x + 144*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] = ((1331 + (4356*n + 4356)*x + (1584*n^2 + 7920*n + 4752)*x^2 + (1728*n^2 + 3456*n + 1728)*x^3)*v[i - 1])/(121 + (396*n - 132)*x + 144*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.