Real-rooted atlas

A034870 → Veronese 2 → basis A013610

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-9\,x^{2}\right)\,P_{n-2}(x) + \left(4 + 6\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + 3\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1235
2824941
33214416227365
41287681440864813281
551238401008010800405024329525
620481843262208967686804017496729265721
78192860163548167257607620483810247144221872391485
83276839321619169284866048684288052254721959552279936656121523361
913107217694729953280301916165337446455427328323326089447840106288219683193710245
10524288786432050135040176947200377395200498161664400308480184757760433026003936600590491743392201
1120971523460300824654643299267379224816844803985293312410983372826420359689907634881905314401428985817714715690529805
128388608150994944118908518453791948801543156531229238755328369545379843074369126416212493440504388684881062467251018336531441141214768241
1333554432654311424564343603228339863552917825126402006103490563009155235843088343531522123236177929367218432024588948384335303841617962705815943231270932914165
141342177282818572288264241152001458611159045264674652161304723718144226844010086427776817561602369709748224137193722265651447645849611555086233613541116680624974616478296911438396227481
Technical details
Node id
n-8a56d090843213ac · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-38d9a649e263b473

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:      2,       3;
  2:      8,      24,       9;
  3:     32,     144,     162,       27;
  4:    128,     768,    1440,      864,       81;
  5:    512,    3840,   10080,    10800,     4050,      243;
  6:   2048,   18432,   62208,    96768,    68040,    17496,      729;
  7:   8192,   86016,  354816,   725760,   762048,   381024,    71442,    2187;
  8:  32768,  393216, 1916928,  4866048,  6842880,  5225472,  1959552,  279936,    6561;
  9: 131072, 1769472, 9953280, 30191616, 53374464, 55427328, 32332608, 9447840, 1062882, 19683;
%F Formula
The row polynomials satisfy P(n,x) = -9*x^2*P(n-2,x) + (4+6*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 3*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + 3*x;
P[n_, x_] := P[n, x] = Expand[-9*x^2*P[n - 2, x] + (4 + 6*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = -9*x^2*v[i - 2] + (4 + 6*x)*v[i - 1]); 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.