Real-rooted atlas

A038221 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(1 - x + 3\,n x + n^{2} x^{2}\right)\,P_{n}(x) = \left(3 + 9\,x + 9\,x^{2} + 3\,x^{3} + 9\,n x + 15\,n x^{2} + 6\,n x^{3} + 3\,n^{2} x^{2} + 3\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
131215
297281162
3273247294321512
481129643745184202512960
524348602187038880303758748104976
6729174969841523328027337515746435721816480
72187612364133431224720191362516533727501411399686158592
865612099521653372587865611481750132269769001692335923253144145349632
91968370858863772922645395262001450892820888101522845349632143489071968300327525120
10590492361960239148451133740803100072505356925286076142104534963202152336055904900071449292327947776
11177147779446887687765467668080146146275029463089044010253786374134464023675696559743085002357826572550916816325867520
1253144125509168315675954187067232065765823751515244579224061522716269376814082130812689511691702000424408782691833004889813529113192681472
1315943238290479611192147467295622048284985236257386817323613405705513217509492915216620338978111399409450055173141738179074359363502727631312487308777111293952
14478296926784626439172516112785601145611969379922534471814176870379953944310505695749121163423728467957550393800579317988249250704103104735572802511312446693610761680255289581076480
Technical details
Node id
n-a5dd62aca1f7b90b · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-d42e6339eab21b9c

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,     12;
  2:     9,     72,      81;
  3:    27,    324,     729,      432;
  4:    81,   1296,    4374,     5184,     2025;
  5:   243,   4860,   21870,    38880,    30375,     8748;
  6:   729,  17496,   98415,   233280,   273375,   157464,    35721;
  7:  2187,  61236,  413343,  1224720,  1913625,  1653372,   750141,   139968;
  8:  6561, 209952, 1653372,  5878656, 11481750, 13226976,  9001692,  3359232,   531441;
  9: 19683, 708588, 6377292, 26453952, 62001450, 89282088, 81015228, 45349632, 14348907, 1968300;
%F Formula
The row polynomials satisfy (1+(3*n-1)*x+n^2*x^2)*P(n,x) = (3+(9*n+9)*x+(3*n^2+15*n+9)*x^2+(3*n^2+6*n+3)*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[((3 + (9*n + 9)*x + (3*n^2 + 15*n + 9)*x^2 + (3*n^2 + 6*n + 3)*x^3)*P[n - 1, x])/(1 + (3*n - 1)*x + 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] = ((3 + (9*n + 9)*x + (3*n^2 + 15*n + 9)*x^2 + (3*n^2 + 6*n + 3)*x^3)*v[i - 1])/(1 + (3*n - 1)*x + 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.