Real-rooted atlas

A013613 → basis A013611 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(7\,n + 24\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 24\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
12424
233611521488
33528241924147269192
432928338688116121613271042859936
52881203951360203212804644864039813120110822520
6242020841489280284497920975421440167215104011466178564122597744
719765032406594944348509952015931883520409677004805618427494432105299968149100618408
8158120256379488614439033114624223046369280764730408960157315969843217978967982088806025134085282421909312
912451970163415397529640984770355228103842529281204450394112033036353667072566337491435525547795834470423776267862016184224464087256
1096848656802988472838404098477035520327878162841601686230551756805781361891737601321454146682880194172854206464016643387503411206340338096537606345509318561040
1174573465736255680453952039447841466880360665979125760216399587475456089032973132759042543799232364544049837699246325760640770418881331204882060334333952016738492574859264216381867762931464
125694701019842147715813196836817985369088037869927808204802596795049705472012464616238586265642735827103724339210465916841728409601794157172867727360205046534042025984014060333762881781764382441692326789127317641346164591328
13431848160671217767467181900833504366685870083829070478385152029538543690399744016204001110162145286481600444064858112190479686519457054724081707568274079744062197448659414548480639745186211121070083988021940017378099211394348400049651712245750788542027525432
14325547074967521451009819855232298493448655933443752489068817448963216419201843527680198499013599486279689074240621690801356831111682131511318937680001468338171962982415238374921556564377602089834274956328828928195413075060851526860811166461432048658677762945000817243602288648204295555941534310576
Technical details
Node id
n-13735c4e8406e5f2 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-0ad613824303cac4

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -1)
  • 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:          0;
  1:         24;
  2:        336,        1152;
  3:       3528,       24192,        41472;
  4:      32928,      338688,      1161216,       1327104;
  5:     288120,     3951360,     20321280,      46448640,       39813120;
  6:    2420208,    41489280,    284497920,     975421440,     1672151040,     1146617856;
  7:   19765032,   406594944,   3485099520,   15931883520,    40967700480,    56184274944,    32105299968;
  8:  158120256,  3794886144,  39033114624,  223046369280,   764730408960,  1573159698432,  1797896798208,   880602513408;
  9: 1245197016, 34153975296, 409847703552, 2810384252928, 12044503941120, 33036353667072, 56633749143552, 55477958344704, 23776267862016;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (7*n+24*n*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 24.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 24;
P[n_, x_] := P[n, x] = Expand[Together[((7*n + 24*n*x)*P[n - 1, x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 24; for(n = 2, N, my(i = n + 1); v[i] = ((7*n + 24*n*x)*v[i - 1])/(n - 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.