Real-rooted atlas

A038255 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(5 + x + n\right)\,P_{n}(x) = \left(36 + 12\,x + x^{2} + 6\,n + n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1718
24814163
3324144211490
4216012962882813773
51425610800324048035128812
69331285536324006480720421218491
7606528653184299376756001134010084911647086
839191044852224261273679833615120018144134456112353145
92519424035271936218350087838208179625627216027216172863192236816
1016124313625194240017635968072783360195955203592512453600388802160701686011319
111027924992177367449613856832006466521602001542404311014465862727128005346026407715084554482
1265303470081233509990410642046976554273280019399564804803701768622028811290752106920071280316884137569208117
13413588643848489451110480178149376461155368961801388160050438868481040802048160123392183474721544400926643744911276825744020
142612138803205790241013765942615777283741646970881614043791365043886848011769069312208160409628021593628540512216216011793643689812034669218547
Technical details
Node id
n-7e1907415114e6d6 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-f25561f72e77d4fd

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:        7,        1;
  2:       48,       14,        1;
  3:      324,      144,       21,       1;
  4:     2160,     1296,      288,      28,       1;
  5:    14256,    10800,     3240,     480,      35,      1;
  6:    93312,    85536,    32400,    6480,     720,     42,     1;
  7:   606528,   653184,   299376,   75600,   11340,   1008,    49,    1;
  8:  3919104,  4852224,  2612736,  798336,  151200,  18144,  1344,   56,  1;
  9: 25194240, 35271936, 21835008, 7838208, 1796256, 272160, 27216, 1728, 63, 1;
%F Formula
The row polynomials satisfy (n+5+x)*P(n,x) = (6*n+36+(n+12)*x+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[((6*n + 36 + (n + 12)*x + x^2)*P[n - 1, x])/(n + 5 + 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] = ((6*n + 36 + (n + 12)*x + x^2)*v[i - 1])/(n + 5 + 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.