Real-rooted atlas

A038327 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(144 - 12\,x + 36\,n x + n^{2} x^{2}\right)\,P_{n}(x) = \left(1728 + 432\,x + 36\,x^{2} + x^{3} + 432\,n x + 60\,n x^{2} + 2\,n x^{3} + 12\,n^{2} x^{2} + n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
112416
2144969249
317281728324163796
4207362764877767682556953
524883241472015552023040150036843648
62985984597196827993605529605400025924912366913
7358318088360755247029248116121601512000108864411664179705812
842998169611466178567524679682229534723628800034836481975686144812591996433
9515978035215479341056116095057924013162496783820800940584967112448331776874810037147122064
1061917364224206391214080174142586880687970713601567641600022574039042133734401327104052488012000121529409237929
1174300837068827243640258562554091274240113515167744029561241600049662885888563305881643794432023094720792000159721447507985556084
128916100448256356644017930243677891434905618162426839040532102348800010216365096961351934115841261279641683140992038016000114998420736169106013180231881
1310699320537907246363722330931252159187622297628333385868902492231073792000199219119390723012881743872327932706816259399895041482624000597991681617408263641961491077447838784
14128391846454886459916195012280327302286267121664432728075088691215494820397056003718756895293446327051662131278703849635847263197061124981616640025115650569057484822145763292822520898382354865457
Technical details
Node id
n-47ed1338ea0157c6 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-16965aba131edac8

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:         12,           4;
  2:        144,          96,           9;
  3:       1728,        1728,         324,         16;
  4:      20736,       27648,        7776,        768,        25;
  5:     248832,      414720,      155520,      23040,      1500,       36;
  6:    2985984,     5971968,     2799360,     552960,     54000,     2592,      49;
  7:   35831808,    83607552,    47029248,   11612160,   1512000,   108864,    4116,     64;
  8:  429981696,  1146617856,   752467968,  222953472,  36288000,  3483648,  197568,   6144,   81;
  9: 5159780352, 15479341056, 11609505792, 4013162496, 783820800, 94058496, 7112448, 331776, 8748, 100;
%F Formula
The row polynomials satisfy (144+(36*n-12)*x+n^2*x^2)*P(n,x) = (1728+(432*n+432)*x+(12*n^2+60*n+36)*x^2+(n^2+2*n+1)*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[((1728 + (432*n + 432)*x + (12*n^2 + 60*n + 36)*x^2 + (n^2 + 2*n + 1)*x^3)*P[n - 1, x])/(144 + (36*n - 12)*x + 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] = ((1728 + (432*n + 432)*x + (12*n^2 + 60*n + 36)*x^2 + (n^2 + 2*n + 1)*x^3)*v[i - 1])/(144 + (36*n - 12)*x + 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

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.