Real-rooted atlas

A027465 → basis A013611 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(8 + 12\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
181220
264192144400
35122304345617288000
4409624576552965529620736160000
53276824576073728011059208294402488323200000
626214423592968847360176947201990656011943936298598464000000
720971522202009699090432247726080371589120334430208167215104358318081280000000
81677721620132659210569646083170893824594542592071345111045350883328229323571242998169625600000000
9134217728181193932810871635968380507258888561413324812842119987212842119987282556485632309586821125159780352512000000000
10107374182416106127360108716359680434865438720114152177664020547391979522568423997440220150628352012383472844804127824281606191736422410240000000000
118589934592141733920768106300440576047835198259201435055947776030136174903296452042623549444843313823744036324853678080181624268390405448728051712743008370688204800000000000
1268719476736123695058124810204842295296510242114764801722067137331204132961129594887232681976791049299162541588488717964882739205811976588492802615389464821767132880358604889161004482564096000000000000
13549755813888107202383708169648214533734453065179935539219899442475827205372849468473344107456989369466881611854840542003218133366956097536151111391300812809066683478048768370909778647449692727444661862410699320537907281920000000000000
1443980465111049235897673318490050002314854454030001388912642228737557292646466862126718779392150439785117253632257896774486720512338489516513820672338489516513820672253867137385365504138472984028381184519273690106429441198323900245606412839184645488641638400000000000000
Technical details
Node id
n-bc910c5f6dde261c · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-f06e51c139cfae05

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:         8,         12;
  2:        64,        192,         144;
  3:       512,       2304,        3456,        1728;
  4:      4096,      24576,       55296,       55296,       20736;
  5:     32768,     245760,      737280,     1105920,      829440,       248832;
  6:    262144,    2359296,     8847360,    17694720,    19906560,     11943936,      2985984;
  7:   2097152,   22020096,    99090432,   247726080,   371589120,    334430208,    167215104,    35831808;
  8:  16777216,  201326592,  1056964608,  3170893824,  5945425920,   7134511104,   5350883328,  2293235712,   429981696;
  9: 134217728, 1811939328, 10871635968, 38050725888, 85614133248, 128421199872, 128421199872, 82556485632, 30958682112, 5159780352;
%F Formula
The row polynomials satisfy P(n,x) = (8+12*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[(8 + 12*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] = (8 + 12*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.