Real-rooted atlas

A027467 → basis A013611 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(3 + x + n\right)\,P_{n}(x) = \left(64 + 32\,x + 4\,x^{2} + 16\,n + 4\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
120424
238416016560
3716846089606412800
4131072114688368645120256288000
52359296262144011468802457602560010246400000
6419430405662310431457280917504014745601228804096140800000
77381975041174405120792723456293601280642252808257536573440163843072000000
8128849018882362232012818790481920845571686423488102404110417924404019226214406553666560000000
922333829939246385646796842520176230422548578304076101451776169114337282466250752226492416117964802621441433600000000
103848290697216893353197568092771293593605669356830720225485783040060881161420811274289152014092861440113246208052428800104857630720000000000
1165970697666560169324790677504196537703464960136064563937280623629251379201984274890752044646185041927086696038407751073792055364812802306867204194304655360000000000
12112589990684262431665934879948804063794976260096314460325543936016327747672473605986840813240321587419912601603061452688588842520176230404133906022402657511014410066329601677721613926400000000000
1319140298416324608585467951558164488233143068786688070439112921841664408798423207116801698085757937254451885953714749441179226220789760198994424758272245672129331202149631131648125627793408436207616067108864294912000000000000
14324259173170675712107185671131417804816393102643628605441536853372840181760986147580905783296457854233991970816158488004074143744415087629717995528254583545528320123818753182924813757639242547210943576670208586263035904187904819202684354566225920000000000000
Technical details
Node id
n-d02bb449e1edccb4 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-76caa1d365acf455

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:           20,            4;
  2:          384,          160,           16;
  3:         7168,         4608,          960,           64;
  4:       131072,       114688,        36864,         5120,         256;
  5:      2359296,      2621440,      1146880,       245760,       25600,        1024;
  6:     41943040,     56623104,     31457280,      9175040,     1474560,      122880,       4096;
  7:    738197504,   1174405120,    792723456,    293601280,    64225280,     8257536,     573440,     16384;
  8:  12884901888,  23622320128,  18790481920,   8455716864,  2348810240,   411041792,   44040192,   2621440,    65536;
  9: 223338299392, 463856467968, 425201762304, 225485783040, 76101451776, 16911433728, 2466250752, 226492416, 11796480, 262144;
%F Formula
The row polynomials satisfy (n+3+x)*P(n,x) = (16*n+64+(4*n+32)*x+4*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[((16*n + 64 + (4*n + 32)*x + 4*x^2)*P[n - 1, x])/(n + 3 + 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] = ((16*n + 64 + (4*n + 32)*x + 4*x^2)*v[i - 1])/(n + 3 + 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.