Real-rooted atlas

A038243 → basis A013611 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(9 - 6\,x + 18\,n x + 4\,n^{2} x^{2}\right)\,P_{n}(x) = \left(54 + 108\,x + 72\,x^{2} + 16\,x^{3} + 108\,n x + 120\,n x^{2} + 32\,n x^{3} + 24\,n^{2} x^{2} + 16\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
161622
236192144372
32161728259210245560
41296138243110424576640077200
57776103680311040368640192000368641020000
646656746496279936044236803456000132710420070413000000
727993652254722351462446448640483840002786918484295681048576161200000
81679616358318081881169924459069445806080004459069442023096325033164853084161956000000
9100776962418647041451188224401316249662705664006019743744364157337613589544962866544642621440023320000000
1060466176161243136010883911680343985356806270566400072236924928546236006402717908992085996339201572864000126877696274000000000
1136279705610642046976798153523202837879193605912248320007946061742087210315284484484549836801891919462405190451200083739279366039797763180000000000
12217678233669657034752574670536704227030335488053210234880008173092077568865237834137664577517649923405455032320124570828800030146140569643486543872283534950436520000000000
13130606940164527707258884074936532992177083661680644611553689600079687647756288964122158039048395077294489653125098504192242913116160007837996548096169597521100822115726131213153337344415600000000000
1478364164096292559545958428524555730944135227523465216387370509926400743751379058688101232826594099210074092753387527437513790586884080940351488001645979275100164748730590822492886049751041104880336896603979776004692000000000000
Technical details
Node id
n-ad2b43f723b42ee4 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-e543af4f498eab8b

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:        6,        16;
  2:       36,       192,        144;
  3:      216,      1728,       2592,       1024;
  4:     1296,     13824,      31104,      24576,       6400;
  5:     7776,    103680,     311040,     368640,     192000,      36864;
  6:    46656,    746496,    2799360,    4423680,    3456000,    1327104,     200704;
  7:   279936,   5225472,   23514624,   46448640,   48384000,   27869184,    8429568,    1048576;
  8:  1679616,  35831808,  188116992,  445906944,  580608000,  445906944,  202309632,   50331648,   5308416;
  9: 10077696, 241864704, 1451188224, 4013162496, 6270566400, 6019743744, 3641573376, 1358954496, 286654464, 26214400;
%F Formula
The row polynomials satisfy (9+(18*n-6)*x+4*n^2*x^2)*P(n,x) = (54+(108*n+108)*x+(24*n^2+120*n+72)*x^2+(16*n^2+32*n+16)*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[((54 + (108*n + 108)*x + (24*n^2 + 120*n + 72)*x^2 + (16*n^2 + 32*n + 16)*x^3)*P[n - 1, x])/(9 + (18*n - 6)*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] = ((54 + (108*n + 108)*x + (24*n^2 + 120*n + 72)*x^2 + (16*n^2 + 32*n + 16)*x^3)*v[i - 1])/(9 + (18*n - 6)*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.