Real-rooted atlas

A038327 → basis A013609 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(13 + 2\,n x\right)\,P_{n}(x) = \left(169 + 52\,x + 4\,x^{2} + 26\,n x + 4\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
113417
216910412285
321972028468324725
428561351521216816648077625
53712935712202636405408052001921265625
648268098911032514098014060802028001497644820503125
76274851713515065293565836319883206151600681408407681024330328125
88157307212007952544162180782466535705615994160023622144211993610649623045296640625
9106044993732936630595627107359344129744625923742633440690947712826775046230016269568512084575390625
101378584918494241799749204404945893402409543052808109039120017964640512268701888026996736017521920665600112641345517578125
11179216039403760657736413566998969586180430705820688016565608488004281572655367684873996896513331208352115204759040016107522457621336064453125
122329808512248186023698913776109183925544408746556755859203230293655160095417904890881998067239168301121593344325732492802474700800125638656383385653248337340478515625
133028751065922531211500426369012167746212881863212616809174020486065773641356002015703240819844823905191705684815915458561100975825664104556108800707764428832396083289989121146885319599853515625
1439373763856992891696100596916616825441508953749252208750842697429761103970802726792040761998869912321097438431113024220521380192256333962667118083805842360320322032815104196536238088189009922087321624576083686387939453125
Technical details
Node id
n-764c27fefb941c66 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-a43792dbb4f60ed3

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:          13,           4;
  2:         169,         104,          12;
  3:        2197,        2028,         468,          32;
  4:       28561,       35152,       12168,        1664,         80;
  5:      371293,      571220,      263640,       54080,       5200,       192;
  6:     4826809,     8911032,     5140980,     1406080,     202800,     14976,      448;
  7:    62748517,   135150652,    93565836,    31988320,    6151600,    681408,    40768,    1024;
  8:   815730721,  2007952544,  1621807824,   665357056,  159941600,  23622144,  2119936,  106496,   2304;
  9: 10604499373, 29366305956, 27107359344, 12974462592, 3742633440, 690947712, 82677504, 6230016, 269568, 5120;
%F Formula
The row polynomials satisfy (13+2*n*x)*P(n,x) = (169+(26*n+52)*x+(4*n+4)*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[((169 + (26*n + 52)*x + (4*n + 4)*x^2)*P[n - 1, x])/(13 + 2*n*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] = ((169 + (26*n + 52)*x + (4*n + 4)*x^2)*v[i - 1])/(13 + 2*n*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

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.