Real-rooted atlas

A013622 → basis A013609 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(16 - 20\,x + 60\,n x + 25\,n^{2} x^{2}\right)\,P_{n}(x) = \left(128 + 480\,x + 600\,x^{2} + 250\,x^{3} + 480\,n x + 1000\,n x^{2} + 500\,n x^{3} + 200\,n^{2} x^{2} + 250\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
184048
2646409001604
35127680216001600045792
44096819203456005120002500001193616
53276881920046080001024000010000000360000029299968
626214478643205529600016384000024000000017280000049000000689062464
7209715273400320619315200229376000044800000004838400000274400000064000000015690972672
8167772166710886406606028800293601280007168000000010321920000087808000000409600000008100000000348421222656
913421772860397977606794772480035232153600010321920000001857945600000210739200000014745600000005832000000001000000000007581732876288
101073741824536870912006794772480004026531840000137625600000002972712960000042147840000000393216000000002332800000000080000000000001210000000000162257899521024
118589934592472446402560664377753600044291850240000173015040000000435997900800000741801984000000865075200000000684288000000000352000000000000106480000000000144000000000003424474788913152
126871947673641231686041606378026434560047244640256000020761804800000005979399782400000118688317440000001660944384000000016422912000000000112640000000000005111040000000000138240000000000016900000000000071423626401386496
135497558138883573412790272060301340835840049134425866240002399141888000000077732197171200000176336928768000000287897026560000000341596569600000000292864000000000000177182720000000000718848000000000001757600000000000019600000000000001474574400857899008
1443980465111043078632557772805628125144678400500277790638080002687038914560000009673340092416000002468717002752000000460635242496000000063764692992000000006560153600000000000496111616000000000026836992000000000009842560000000000002195200000000000002250000000000000030174789753120374784
Technical details
Node id
n-2de0c804fb24112d · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-51a5affbafca912b

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:         8,         40;
  2:        64,        640,         900;
  3:       512,       7680,       21600,        16000;
  4:      4096,      81920,      345600,       512000,        250000;
  5:     32768,     819200,     4608000,     10240000,      10000000,       3600000;
  6:    262144,    7864320,    55296000,    163840000,     240000000,     172800000,      49000000;
  7:   2097152,   73400320,   619315200,   2293760000,    4480000000,    4838400000,    2744000000,     640000000;
  8:  16777216,  671088640,  6606028800,  29360128000,   71680000000,  103219200000,   87808000000,   40960000000,   8100000000;
  9: 134217728, 6039797760, 67947724800, 352321536000, 1032192000000, 1857945600000, 2107392000000, 1474560000000, 583200000000, 100000000000;
%F Formula
The row polynomials satisfy (16+(60*n-20)*x+25*n^2*x^2)*P(n,x) = (128+(480*n+480)*x+(200*n^2+1000*n+600)*x^2+(250*n^2+500*n+250)*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[((128 + (480*n + 480)*x + (200*n^2 + 1000*n + 600)*x^2 + (250*n^2 + 500*n + 250)*x^3)*P[n - 1, x])/(16 + (60*n - 20)*x + 25*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] = ((128 + (480*n + 480)*x + (200*n^2 + 1000*n + 600)*x^2 + (250*n^2 + 500*n + 250)*x^3)*v[i - 1])/(16 + (60*n - 20)*x + 25*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.