Real-rooted atlas

A027466 → basis A013610 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(5 + 3\,x + 3\,n\right)\,P_{n}(x) = \left(64 + 48\,x + 9\,x^{2} + 24\,n + 9\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
111314
2112669187
310881008297272420
41024013056604811888130613
59420815360097920302404455243380666
685196816957441382400587520136080160387294670479
7760217617891328178053129676800308448057153656133218756689952
867108864182452224214695936142442496580608001480550422861441924566561682050985
9587202560181193932824631050241932263424961486848313528320666247688817984649539196838145637478
10510027366417616076800271790899202463105024014491975680576892108815676416002855347203306744021651305904996675855331
114402341478416830903091229066526720029896998912020320616448095647039488317290659847390310400117783072012124728071449291771471141246682444
12377957122048158484293222430295625564163487983206400269072990208014630843842565738822369281631780536323325639680047113228804364902082338340453144113409648518717
133229815406592147403277598723090443717836839384313233408340078362624002098769323622495100484976643197343891456795493011456144111052800183741592321547556192759960631594323156921418836050
14274877906944001356522470768643095468829573124326621204971524135352889507842856658246041601469138526535685706029098598416786055430144371230072012860526642176070155880704541644667224552574247829691829703743628343
Technical details
Node id
n-f44c764f0d48352a · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-bd3dad0f6f053f13

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,          3;
  2:       112,         66,          9;
  3:      1088,       1008,        297,         27;
  4:     10240,      13056,       6048,       1188,        81;
  5:     94208,     153600,      97920,      30240,      4455,       243;
  6:    851968,    1695744,    1382400,     587520,    136080,     16038,      729;
  7:   7602176,   17891328,   17805312,    9676800,   3084480,    571536,    56133,    2187;
  8:  67108864,  182452224,  214695936,  142442496,  58060800,  14805504,  2286144,  192456,   6561;
  9: 587202560, 1811939328, 2463105024, 1932263424, 961486848, 313528320, 66624768, 8817984, 649539, 19683;
%F Formula
The row polynomials satisfy (3*n+5+3*x)*P(n,x) = (24*n+64+(9*n+48)*x+9*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[((24*n + 64 + (9*n + 48)*x + 9*x^2)*P[n - 1, x])/(3*n + 5 + 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] = ((24*n + 64 + (9*n + 48)*x + 9*x^2)*v[i - 1])/(3*n + 5 + 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.