Real-rooted atlas

A013620 → basis A013611 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(17 + 24\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1172441
22898165761681
3491320808293761382468921
4835214716489987849400323317762825761
51419857100225202829888039951360282009607962624115856201
624137569204459408721621440135834624014382489608121876481911029764750104241
7410338673405511159217174590272404108006405705054208048325165056227412541444586471424194754273881
869757574417878502521638929071283210991737774081939718430720219074081587215464052817926237601136641100753141767984925229121
9118587876497150676360725685087827233282802893132390459355383980032837958362071047886666937139247717648695296168415230689282641807540224327381934393961
10201599390044928461090359280180811632870720680702617866240168173587943424028490584310415363351833448284160270400009273344014315294608588804491072818380806340338096537613422659310152401
113427189630763353222238971853637568639274249601591142369262336044926372779171840887956544341278721253585709658275841264120043352883208923200306020352041991530851860480118564322405253121521681143169024550329031716248441
1258262223722976198703061365983047664002411946918436065893703279616011456225058688819202587759072080297984426219141283813785651576097768797634564550832156070379520285542409792651264012093560885335818243104229532064808963652034743605657622563490300366186081
13990457803290593718177813801568543215397677573093354247970562508424795136281313970885581004807148684436621823180813456347174817549516818997196011507128729620114678129831077478415776218141043982336089089231855307194368343017363293161390088070996783368503296876488338465357824925103102315013629321
1416837782655940092933279382190563948323053872718663515257617245398881864556748866952725070768279142418904298843511043522564003263284508220981248645904664391242376806479788223248329940664327509479835136935591936530080929539077806489627212710821257470279689604486172208518922242086042245547551621122103572012316858777637929227194915558802161
Technical details
Node id
n-f8421f7ef3e8d09d · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-3d4815a526d230d8

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:           17,            24;
  2:          289,           816,           576;
  3:         4913,         20808,         29376,          13824;
  4:        83521,        471648,        998784,         940032,         331776;
  5:      1419857,      10022520,      28298880,       39951360,       28200960,        7962624;
  6:     24137569,     204459408,     721621440,     1358346240,     1438248960,      812187648,      191102976;
  7:    410338673,    4055111592,   17174590272,    40410800640,    57050542080,    48325165056,    22741254144,     4586471424;
  8:   6975757441,   78785025216,  389290712832,  1099173777408,  1939718430720,  2190740815872,  1546405281792,   623760113664,   110075314176;
  9: 118587876497, 1506763607256, 8508782723328, 28028931323904, 59355383980032, 83795836207104, 78866669371392, 47717648695296, 16841523068928, 2641807540224;
%F Formula
The row polynomials satisfy P(n,x) = (17+24*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[(17 + 24*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] = (17 + 24*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

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.