Real-rooted atlas

A038271 → basis A007318 → basis A007318

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(17 + 5\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
117522
228917025484
349134335127512510648
48352198260433508500625234256
51419857208802512282503612505312531255153632
624137569425957103132037512282500270937531875015625113379904
7410338673844814915745424925365404375107471875189656251859375781252494357888
869757574411641354692016896298300993899900036540437508597750001264375001062500039062554875873536
91185878764973139090848453693048057002534444745001118137387503288639375064483125008128125005976562519531251207269217792
10201599390044959293938248507847727121125615508009500031680559312501118137387500274053281250460593750005080078125332031250976562526559922791424
1134271896307633110879664524695163058330183375143874997220625846323513062503484861524375010249592718750215327578125031665820312531044921875182617187548828125584318301411328
125826222372297612056313778457980332638993574085032611666036675002158124958309375101558821567500034848615243750087853651875000161495683593752111054687500186269531250996093750024414062512855002631049216
1399045780329059373787044541993446566830197799884350720717819410517505299395730959687528055624458021875110022056698125003235942844062500713810921484375116635771484375137218554687501100683593750539550781251220703125282810057883082752
1416837782655940092969332046230341559013254655896977062751559371281997301500126125618396840562574191540233435625032731561867692187511002205669812500028314499885546875555186272265625081645040039062587320898437500642065429687529052734375061035156256221821273427820544
Technical details
Node id
n-f941cf008016de0c · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-75fb8f2b619cd489

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,            5;
  2:          289,          170,           25;
  3:         4913,         4335,         1275,          125;
  4:        83521,        98260,        43350,         8500,          625;
  5:      1419857,      2088025,      1228250,       361250,        53125,        3125;
  6:     24137569,     42595710,     31320375,     12282500,      2709375,      318750,      15625;
  7:    410338673,    844814915,    745424925,    365404375,    107471875,    18965625,    1859375,     78125;
  8:   6975757441,  16413546920,  16896298300,   9938999000,   3654043750,   859775000,  126437500,  10625000,   390625;
  9: 118587876497, 313909084845, 369304805700, 253444474500, 111813738750, 32886393750, 6448312500, 812812500, 59765625, 1953125;
%F Formula
The row polynomials satisfy P(n,x) = (17+5*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 + 5*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 + 5*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.