Real-rooted atlas

A013610 → basis A013611 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-2 + 3\,x + 3\,n\right)\,P_{n}(x) = \left(4 + 24\,x + 36\,x^{2} + 12\,n + 36\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1161228
2112384144640
364040326912172813312
43328307209676811059220736262144
516384199680921600193536016588802488324980736
67782411796487188480221184003483648023887872298598492274688
7360448653721649545216201277440464486400585252864334430208358318081677721600

All 15 rows

Technical details
Node id
n-630a3984525684cf · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-1f45c55ca90e4ac8

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:       1;
  1:      16,        12;
  2:     112,       384,        144;
  3:     640,      4032,       6912,        1728;
  4:    3328,     30720,      96768,      110592,       20736;
  5:   16384,    199680,     921600,     1935360,     1658880,       248832;
  6:   77824,   1179648,    7188480,    22118400,    34836480,     23887872,      2985984;
  7:  360448,   6537216,   49545216,   201277440,   464486400,    585252864,    334430208,     35831808;
  8: 1638400,  34603008,  313786368,  1585446912,  4830658560,   8918138880,   9364045824,   4586471424,   429981696;
  9: 7340032, 176947200, 1868562432, 11296309248, 42807066624, 104342224896, 160526499840, 144473849856, 61917364224, 5159780352;
%F Formula
The row polynomials satisfy (3*n-2+3*x)*P(n,x) = (12*n+4+(36*n+24)*x+36*x^2)*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[((12*n + 4 + (36*n + 24)*x + 36*x^2)*P[n - 1, x])/(3*n - 2 + 3*x)]];
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] = ((12*n + 4 + (36*n + 24)*x + 36*x^2)*v[i - 1])/(3*n - 2 + 3*x)); 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.