Real-rooted atlas

A013610 → basis A007318 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(7 + 9\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
17916
24912681256
3343132317017294096
42401123482381420412656165536
516807108045277830357210229635590491048576
6117649907578291721550009404822335248005853144116777216
7823543741188728588707612615157876480560761421260406094782969268435456
857648015929509626682793268612896811027072701134213192729137052267846264430467214294967296
940353607466948881240145138872043541641389411160217863857774153118780928437157316271194342338742048968719476736
1028247524936318246302101269964572043541640162097968690250094008836267957866610196867004040949180198052711943423034867844011099511627776
111977326743279650496511797753191856934190882851783077655590320953978006241265511457943789689827770243622917499510440982178552684823988773138105960917592186044416
121384128720121355128824415101126811546471911490660187223153836953851447736074457771716041116636667891065365116081267489529234750099940112762607528342636009007156282429536481281474976710656
1396889010407161943060251712492750362274588943945650061893034111018054381021799784637510323085345089656129681157969311267906830896650905647736353420465761692981199384098255982570108781977125418658283294503599627370496
14678223072849122080153112821020241279585715246955152155081855173428797689477044595976548692001457795477231351858155362114415208404247823787130357750695632468380141116147801391798805430286812593533031687792491028511762422287679245496172057594037927936
Technical details
Node id
n-2b84091cce44a505 · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-27210632b0078bca

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:        7,         9;
  2:       49,       126,         81;
  3:      343,      1323,       1701,        729;
  4:     2401,     12348,      23814,      20412,        6561;
  5:    16807,    108045,     277830,     357210,      229635,       59049;
  6:   117649,    907578,    2917215,    5000940,     4822335,     2480058,      531441;
  7:   823543,   7411887,   28588707,   61261515,    78764805,    60761421,    26040609,    4782969;
  8:  5764801,  59295096,  266827932,  686128968,  1102707270,  1134213192,   729137052,  267846264,   43046721;
  9: 40353607, 466948881, 2401451388, 7204354164, 13894111602, 17863857774, 15311878092, 8437157316, 2711943423, 387420489;
%F Formula
The row polynomials satisfy P(n,x) = (7+9*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[(7 + 9*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] = (7 + 9*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

Candidate equalities ?

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.