Real-rooted atlas

A038243 → basis A013611 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + 2\,x + 2\,n\right)\,P_{n}(x) = \left(18 + 24\,x + 8\,x^{2} + 12\,n + 8\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
110414
2848016180
36481008480642200
44752103688064256025626000
5336969504010368053760128001024300000
623328080870411404808294403225606144040963400000
7158630465318401132185610644480580608018063362867201638438000000
8106375685076172810450944012076646485155840371589129633792131072065536420000000
970543872382952448913711104125411328010868981766131220482229534724954521658982402621444600000000
1046357401628217548807659048960121828147201254113280086951854084087480320127401984024772608026214400104857650000000000
11302330880020397256704620786073601123327180801340109619201103619686406376469299225692733440700710912012111052801153433604194304540000000000
121959104102414511882240048953416089699325771776013479926169601286505234432882895749120437243609088154156400640373712486405813305344503316480167772165800000000000
13126253375488101873413324837730893824008485258788864129123503308801401912321638411149712031744655865413632028420834590728906814259201943304929282748107980821810380806710886462000000000000
148097630289927070189027328285245557309447043100180480011879362304409614461832370585613084515001958489197696253952459105789542401768407485644849878159851529893188730881282450391049395240960268435456660000000000000
Technical details
Node id
n-b4f61ec243c292b0 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-7ad822d7676abaf6

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:       10,         4;
  2:       84,        80,        16;
  3:      648,      1008,       480,         64;
  4:     4752,     10368,      8064,       2560,        256;
  5:    33696,     95040,    103680,      53760,      12800,      1024;
  6:   233280,    808704,   1140480,     829440,     322560,     61440,      4096;
  7:  1586304,   6531840,  11321856,   10644480,    5806080,   1806336,    286720,    16384;
  8: 10637568,  50761728, 104509440,  120766464,   85155840,  37158912,   9633792,  1310720,   65536;
  9: 70543872, 382952448, 913711104, 1254113280, 1086898176, 613122048, 222953472, 49545216, 5898240, 262144;
%F Formula
The row polynomials satisfy (2*n+1+2*x)*P(n,x) = (12*n+18+(8*n+24)*x+8*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 + 18 + (8*n + 24)*x + 8*x^2)*P[n - 1, x])/(2*n + 1 + 2*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 + 18 + (8*n + 24)*x + 8*x^2)*v[i - 1])/(2*n + 1 + 2*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.