Real-rooted atlas

A038255 → basis A013611 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(3 + 4\,x + 4\,n\right)\,P_{n}(x) = \left(49 + 56\,x + 16\,x^{2} + 28\,n + 16\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
111415
21058816209
39311260528642783
478891489610080281625635937
56482715778014896067200140801024453871
65210171555848189336011916804032006758440965636785
74117715145884762178187217671360834176022579203153921638469090879
8321181771317668802334156162323399681413708805338726412042240144179265536837948353
92478864431156254372237180384028009873922091059712101787033632032358461931520648806426214410074867407
10189661952999154577202312508744031624051200280098739201672847769667858022401830420480309657600288358401048576120255332241
1114406237699834512592762181400698403391679491203478645632002464868904961226755031044265361408010067312640151388160012687769641943041426558353055
12108752970865691499409552200283022262434902411174404070015389440333949980672019718951239688412034498562559216844805369233408072666316805536481281677721616833388566049
1381663594485956551544849801797898464835234715723858816453731345267204232816005017628942331658240146483637780485467822424064147865862144027920013721634351349760239914188867108864197720987733423
146104007655641457316129121041583443255794403356077134359044860201340234245081791066992643950628271349762315386532659201025385464463363402200619417682804882800641421382516736160306298880103347650562684354562313021713643377
Technical details
Node id
n-f9842bbe09a0e52e · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-c09fbe1bbbcc2fe2

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:        11,          4;
  2:       105,         88,         16;
  3:       931,       1260,        528,         64;
  4:      7889,      14896,      10080,       2816,        256;
  5:     64827,     157780,     148960,      67200,      14080,       1024;
  6:    521017,    1555848,    1893360,    1191680,     403200,      67584,      4096;
  7:   4117715,   14588476,   21781872,   17671360,    8341760,    2257920,    315392,    16384;
  8:  32118177,  131766880,  233415616,  232339968,  141370880,   53387264,  12042240,  1441792,   65536;
  9: 247886443, 1156254372, 2371803840, 2800987392, 2091059712, 1017870336, 320323584, 61931520, 6488064, 262144;
%F Formula
The row polynomials satisfy (4*n+3+4*x)*P(n,x) = (28*n+49+(16*n+56)*x+16*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[((28*n + 49 + (16*n + 56)*x + 16*x^2)*P[n - 1, x])/(4*n + 3 + 4*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] = ((28*n + 49 + (16*n + 56)*x + 16*x^2)*v[i - 1])/(4*n + 3 + 4*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.