Real-rooted atlas

A038220 → basis A007318 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(5\,n + 2\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
122
220828
315012024294
410001200480642744
56250100006000160016024010
6375007500060000240004800384201684
721875052500052500028000084000134408961647086
81250000350000042000002800000112000026880035840204813176688
97031250225000003150000025200000126000004032000806400921604608103766418
10390625001406250002250000002100000001260000005040000013440000230400023040010240807072140
11214843750859375000154687500016500000001155000000554400000184800000422400006336000563200225286214455478
121171875000515625000010312500000123750000009900000000554400000022176000006336000001267200001689600013516804915247455841832
13634765625030468750000670312500008937500000080437500000514800000002402400000082368000002059200000366080000439296003194880106496359873467226
143417968750017773437500042656250000062562500000062562500000045045000000024024000000096096000000288288000006406400000102502400011182080074547202293762712892291396
Technical details
Node id
n-699392d99654cde6 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-9629d1e9c04dbfbe

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:       2;
  2:      20,        8;
  3:     150,      120,       24;
  4:    1000,     1200,      480,       64;
  5:    6250,    10000,     6000,     1600,      160;
  6:   37500,    75000,    60000,    24000,     4800,     384;
  7:  218750,   525000,   525000,   280000,    84000,   13440,    896;
  8: 1250000,  3500000,  4200000,  2800000,  1120000,  268800,  35840,  2048;
  9: 7031250, 22500000, 31500000, 25200000, 12600000, 4032000, 806400, 92160, 4608;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (5*n+2*n*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((5*n + 2*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] = 2; for(n = 2, N, my(i = n + 1); v[i] = ((5*n + 2*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.