Real-rooted atlas

A038259 → basis A013610 → x ↦ x+1

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(P_{n}(x) = \left(26 + 15\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1261541
26767802251681
3175763042017550337568921
44569761054560912600351000506252825761
5118813763427320039546000228150006581250759375115856201
6308915776106932384015422940001186380000513337500118462500113906254750104241
7803181017632436156480561395016005398029000031142475000107800875002073093750170859375194754273881
82088270645769638172211201946169388800224558006400016194087000007474194000002156017500003553875000025628906257984925229121
95429503678976281916537177606505766242560087577622496000757883271600004372403490000016816936500000415803375000059971640625038443359375327381934393961
101411670956533768144255518464002114374028832000325288312128000032841608436000002273649814800000109310087250000036036292500000077963132812500999527343750057665039062513422659310152401
11367034448698777623292570782807040671901080273280001162905715857600001341814287528000001083773078388000006252536990700000025765949137500000743248532812500014293241015625001649220117187508649755859375550329031716248441
129542895666168217666066200765779968020963313704526336004031406481639680000523307572135920000048305314351008000003251319235164000000160779522618000000057973385559375000014864970656250000025727833828125000269872382812500012974633789062522563490300366186081
13248115287320373657618608646549028024320644145457466354688001362615390794211840001965310659799344000002040899531330088000001569922716407760000009057246440814000000039190008638137500000125609002045312500002898669277968750000456084326953125000438542622070312501946195068359375925103102315013629321
146450997470329715097652104210337278468096019539078876479425536004509018202264482816000715373080166961216000082543047711572448000007143148359655308000000470976814922328000000023775271907136750000009144335348898750000002637789042951562500005533823167031250000079814757216796875007084150048828125002919292602539062537929227194915558802161
Technical details
Node id
n-f8abeed5fe1a08b5 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-f7b190274eded71d

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:            26,             15;
  2:           676,            780,            225;
  3:         17576,          30420,          17550,           3375;
  4:        456976,        1054560,         912600,         351000,          50625;
  5:      11881376,       34273200,       39546000,       22815000,        6581250,         759375;
  6:     308915776,     1069323840,     1542294000,     1186380000,      513337500,      118462500,       11390625;
  7:    8031810176,    32436156480,    56139501600,    53980290000,    31142475000,    10780087500,     2073093750,     170859375;
  8:  208827064576,   963817221120,  1946169388800,  2245580064000,  1619408700000,   747419400000,   215601750000,   35538750000,   2562890625;
  9: 5429503678976, 28191653717760, 65057662425600, 87577622496000, 75788327160000, 43724034900000, 16816936500000, 4158033750000, 599716406250, 38443359375;
%F Formula
The row polynomials satisfy P(n,x) = (26+15*x)*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[(26 + 15*x)*P[n - 1, 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] = (26 + 15*x)*v[i - 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

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.