Real-rooted atlas

A130123 → basis A013609 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + 2\,x + 2\,n\right)\,P_{n}(x) = \left(2 + 8\,x + 8\,x^{2} + 4\,n + 8\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
16410
220481684
35624028864648
4144896192015362564752
535228808960128007680102433696
68328448345607168076800368644096233280
7192023296118272322560501760430080172032163841586304
843526144037273612615682580480321126422937607864326553610637568
997281566721105920447283211354112185794561926758411796480353894426214470543872
1021504389120313344014745600447283209083289612386304011010048058982400157286401048576463574016
11471049461768560640459571201622016003936092166661079047785676806055526402883584006920601641943043023308800
121024002260992227082241369702405514854401557135360314887372845675970564671406080322961408013841203203019898881677721619591041024
132211845324800587857923936092161780613120573544857613495173120233916334082968938086426990346240167939932166543114240130862284867108864126253375488
144751361238630414909440010973347845510529024199428669445353085337610796138496016374143385618473392537615114593894485496692736305345331205637144576268435456809763028992
Technical details
Node id
n-7e97a6745522f0af · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-4be2d3b20c0d29e0

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:    6,      4;
  2:   20,     48,      16;
  3:   56,    240,     288,      64;
  4:  144,    896,    1920,    1536,      256;
  5:  352,   2880,    8960,   12800,     7680,     1024;
  6:  832,   8448,   34560,   71680,    76800,    36864,     4096;
  7: 1920,  23296,  118272,  322560,   501760,   430080,   172032,    16384;
  8: 4352,  61440,  372736, 1261568,  2580480,  3211264,  2293760,   786432,   65536;
  9: 9728, 156672, 1105920, 4472832, 11354112, 18579456, 19267584, 11796480, 3538944, 262144;
%F Formula
The row polynomials satisfy (2*n-1+2*x)*P(n,x) = (4*n+2+(8*n+8)*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[((4*n + 2 + (8*n + 8)*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] = ((4*n + 2 + (8*n + 8)*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.