Real-rooted atlas

A038291 → basis A013611 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(3 + 2\,x + 2\,n\right)\,P_{n}(x) = \left(50 + 40\,x + 8\,x^{2} + 20\,n + 8\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
114418
218011216308
322002160672645096
4260003520017280358425682320
53000005200003520001152001792010241306144
6340000072000006240000281600069120086016409620437312
738000000952000001008000005824000019712000387072040140816384316240512
8420000000121600000015232000001075200000465920000126156800206438401835008655364849021184
9460000000015120000000218880000001827840000096768000003354624000756940800106168320825753626214473789452800
1050000000000184000000000302400000000291840000000182784000000774144000002236416000043253760005308416003670016010485761115696526336
1154000000000022000000000004048000000000443520000000032102400000001608499200000567705600000140574720000237895680002595225600161480704419430416776769988608
12580000000000025920000000000528000000000006476800000000053222400000000308183040000001286799360000038928384000008434483200001268776960001245708288070464307216777216251073040519168
136200000000000030160000000000067392000000000091520000000000084198400000000055351296000000026709196800000095590809600000253034496000004873256960000659764019200588880281603053453312671088643741798216769536
146600000000000003472000000000000844480000000000012579840000000000128128000000000009430220800000000516612096000000021367357440000006691356672000001574436864000002729023897600033587986432002748107980801315333734426843545655560034127790080
Technical details
Node id
n-665db32c3a2797ef · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-b29ddf49d7bb6a39

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:         14,           4;
  2:        180,         112,          16;
  3:       2200,        2160,         672,          64;
  4:      26000,       35200,       17280,        3584,        256;
  5:     300000,      520000,      352000,      115200,      17920,       1024;
  6:    3400000,     7200000,     6240000,     2816000,     691200,      86016,      4096;
  7:   38000000,    95200000,   100800000,    58240000,   19712000,    3870720,    401408,     16384;
  8:  420000000,  1216000000,  1523200000,  1075200000,  465920000,  126156800,  20643840,   1835008,   65536;
  9: 4600000000, 15120000000, 21888000000, 18278400000, 9676800000, 3354624000, 756940800, 106168320, 8257536, 262144;
%F Formula
The row polynomials satisfy (2*n+3+2*x)*P(n,x) = (20*n+50+(8*n+40)*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[((20*n + 50 + (8*n + 40)*x + 8*x^2)*P[n - 1, x])/(2*n + 3 + 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] = ((20*n + 50 + (8*n + 40)*x + 8*x^2)*v[i - 1])/(2*n + 3 + 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.