Real-rooted atlas

A038243 → basis A013609 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(2 + x + n\right)\,P_{n}(x) = \left(18 + 12\,x + 2\,x^{2} + 6\,n + 2\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
18210
26032496
3432360968896
4302434561440256168192
520736302401728048006403273728
61399682488321814406912014400153664655360
7933120195955217418248467202419204032035841285767168
86158592149299201567641692897283386880774144107520819225650331648
940310784110854656134369280940584964180377612192768232243227648018432512436207616
1026202009680621568011085465608957952004702924801672151044064256066355206912004096010243758096384
1116930529285764442112886837248081293414404926873600206928691261312204812773376018247680168960090112204832212254720
12108839116804063327027269173305344709469798404877604864023648993280827714764821021327363832012804866048040550401966084096274877906944
13696570347522829817036805282325135365995019796484611553689602536354529281024789708803074369126468319313921107025920126517248958464042598481922336462209024
14444063596544195039697305639617438515204930170126336419651385753625824700661761183632113664409915883520107602919424212548976643099672576322043904223641609175041638419791209299968
Technical details
Node id
n-996c7ba182fc6317 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-966922252c303ef1

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:        8,         2;
  2:       60,        32,         4;
  3:      432,       360,        96,        8;
  4:     3024,      3456,      1440,      256,       16;
  5:    20736,     30240,     17280,     4800,      640,       32;
  6:   139968,    248832,    181440,    69120,    14400,     1536,      64;
  7:   933120,   1959552,   1741824,   846720,   241920,    40320,    3584,    128;
  8:  6158592,  14929920,  15676416,  9289728,  3386880,   774144,  107520,   8192,   256;
  9: 40310784, 110854656, 134369280, 94058496, 41803776, 12192768, 2322432, 276480, 18432, 512;
%F Formula
The row polynomials satisfy (n+2+x)*P(n,x) = (6*n+18+(2*n+12)*x+2*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 + 18 + (2*n + 12)*x + 2*x^2)*P[n - 1, x])/(n + 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] = ((6*n + 18 + (2*n + 12)*x + 2*x^2)*v[i - 1])/(n + 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.