Real-rooted atlas

A013620 → basis A007318 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(5\,n + 3\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 3\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
133
2301848
322527081576
41500270016203246144
5937522500202508100121561440
656250168750202500121500364504374589824
7328125118125017718751417500637875153090153095505024
8187500078750001417500014175000850500030618006123605248850331648
910546875506250001063125001275750009568125045927000137781002361960177147452984832
105859375031640625075937500010631250009568125005740875002296350005904900088573505904904026531840
11322265625193359375052207031258353125000877078125063149625003157481250108256500024357712532476950194861735433480192
1217578125001160156250034804687500626484375007517812500063149625000378897750001623847500048715425009743085001169170206377292309237645312
1395214843756855468750022623046875045246093750061082226562558638937500041047256250021110017500079162565625211100175003799803150414523980207261992680059592704
14512695312503999023437501439648437500316722656250047508398437505130907031250410472562500024628353750001108275918750369425306250886620735001450833930014508339306696156623089744183296
Technical details
Node id
n-6a76f32847485778 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-b1029f17d1b64336

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -1)
  • 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:        0;
  1:        3;
  2:       30,       18;
  3:      225,      270,        81;
  4:     1500,     2700,      1620,       324;
  5:     9375,    22500,     20250,      8100,     1215;
  6:    56250,   168750,    202500,    121500,    36450,     4374;
  7:   328125,  1181250,   1771875,   1417500,   637875,   153090,    15309;
  8:  1875000,  7875000,  14175000,  14175000,  8505000,  3061800,   612360,   52488;
  9: 10546875, 50625000, 106312500, 127575000, 95681250, 45927000, 13778100, 2361960, 177147;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (5*n+3*n*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 3.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 3;
P[n_, x_] := P[n, x] = Expand[Together[((5*n + 3*n*x)*P[n - 1, x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 3; for(n = 2, N, my(i = n + 1); v[i] = ((5*n + 3*n*x)*v[i - 1])/(n - 1)); 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.