Real-rooted atlas

A013610 → basis A013611 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(12 + 4\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
112416
21449616256
317281728576644096
4207362764813824307225665536
5248832414720276480921601536010241048576
6298598459719684976640221184055296073728409616777216
73583180883607552836075524644864015482880309657634406416384268435456
84299816961146617856133772083289181388837158912099090432165150721572864655364294967296
951597803521547934105620639121408160526499848026324992267544166459454259284934656707788826214468719476736
1061917364224206391214080309586821120275188285440160526499840642105999361783627776033973862404246732803145728010485761099511627776
1174300837068827243640258564540606709760454060670976030270711398401412633198592470877732864112113745920186856243202076180480138412032419430417592186044416
128916100448256356644017930246538473662054472649707356160544872805171202905988294246411301065588736322887588249667268247552099656663040996566630460397977616777216281474976710656
13106993205379072463637223309312927274446618624113333543475609694444619563008056666771737804825185231883468883950772944896209876932362243886609858560518214647808471104225282617245696671088644503599627370496
141283918464548864599161950122803212981842252660736173091230035476481586669608658534410577797391056896528889869552844820148185506775045876554106142721305900912476162176501520793626381836615682198486384641127428915226843545672057594037927936
Technical details
Node id
n-fcd6433895993258 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-f9762973ec1089d7

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:         12,           4;
  2:        144,          96,          16;
  3:       1728,        1728,         576,          64;
  4:      20736,       27648,       13824,        3072,        256;
  5:     248832,      414720,      276480,       92160,      15360,       1024;
  6:    2985984,     5971968,     4976640,     2211840,     552960,      73728,      4096;
  7:   35831808,    83607552,    83607552,    46448640,   15482880,    3096576,    344064,    16384;
  8:  429981696,  1146617856,  1337720832,   891813888,  371589120,   99090432,  16515072,  1572864,   65536;
  9: 5159780352, 15479341056, 20639121408, 16052649984, 8026324992, 2675441664, 594542592, 84934656, 7077888, 262144;
%F Formula
The row polynomials satisfy P(n,x) = (12+4*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[(12 + 4*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] = (12 + 4*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.