Real-rooted atlas

A038255 → basis A013610 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(49 - 21\,x + 63\,n x + 9\,n^{2} x^{2}\right)\,P_{n}(x) = \left(343 + 441\,x + 189\,x^{2} + 27\,x^{3} + 441\,n x + 315\,n x^{2} + 54\,n x^{3} + 63\,n^{2} x^{2} + 27\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
171219
24916881298
3343176417014324240
42401164642381412096202556800
516807144060277830211680708758748730000
61176491210104291721529635201488375367416357219100000
7823543988251628588707363031202431012590016921750329139968110800000
85764801790601282668279324065949443403417501680315844900921278382085314411324000000
9403536076225985082401451388426924691242883060502646497448102919345224690355233480783196830015580000000
1028247524948424328402101269964542692469120500302372503705096427218010885410576108288011718274051377810007144929181000000000
11197732674337286732868179775319185410915015280550332609750475487374824277367635314110900845440300769033955304568500550159533255091682080000000000
12138412872012847350509921510112681154383520680928057784924023755705848497888388314689439618631342033926316149712951485279180002310670038621427701128981352923680000000000
13968890104072159240803356124927503622743490038196444858426978735125649040266634765048090962714828257535418112114953924775693379010134500700903245042974960400968173031139312487308267400000000000
1467822307284916277353748376102024127958571310930675683264572584391604225706732734780072618391142932563395605495853568187758077133627662285986362001717212950352931848706431364004785258113062375618410761680252998000000000000
Technical details
Node id
n-1c7b9e2f06debce3 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-fa8d58a7053642bf

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:        1;
  1:        7,        12;
  2:       49,       168,         81;
  3:      343,      1764,       1701,        432;
  4:     2401,     16464,      23814,      12096,       2025;
  5:    16807,    144060,     277830,     211680,      70875,       8748;
  6:   117649,   1210104,    2917215,    2963520,    1488375,     367416,      35721;
  7:   823543,   9882516,   28588707,   36303120,   24310125,    9001692,    1750329,    139968;
  8:  5764801,  79060128,  266827932,  406594944,  340341750,  168031584,   49009212,   7838208,   531441;
  9: 40353607, 622598508, 2401451388, 4269246912, 4288306050, 2646497448, 1029193452, 246903552, 33480783, 1968300;
%F Formula
The row polynomials satisfy (49+(63*n-21)*x+9*n^2*x^2)*P(n,x) = (343+(441*n+441)*x+(63*n^2+315*n+189)*x^2+(27*n^2+54*n+27)*x^3)*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[((343 + (441*n + 441)*x + (63*n^2 + 315*n + 189)*x^2 + (27*n^2 + 54*n + 27)*x^3)*P[n - 1, x])/(49 + (63*n - 21)*x + 9*n^2*x^2)]];
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] = ((343 + (441*n + 441)*x + (63*n^2 + 315*n + 189)*x^2 + (27*n^2 + 54*n + 27)*x^3)*v[i - 1])/(49 + (63*n - 21)*x + 9*n^2*x^2)); 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.