Real-rooted atlas

A038291 → basis A013611 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(25 - 10\,x + 30\,n x + 4\,n^{2} x^{2}\right)\,P_{n}(x) = \left(250 + 300\,x + 120\,x^{2} + 16\,x^{3} + 300\,n x + 200\,n x^{2} + 32\,n x^{3} + 40\,n^{2} x^{2} + 16\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1101626
2100320144564
3100048004320102411144
4100006400086400409606400207760
510000080000014400001024000320000368643720864
61000000960000021600000204800009600000221184020070464692544
710000000112000000302400000358400000224000000774144001404928010485761099312256
81000000001280000000403200000057344000004480000000206438400056197120083886080530841618341949696
91000000000144000000005184000000086016000000806400000004644864000016859136000377487360047775744026214400301482621440
1010000000000160000000000648000000000122880000000013440000000009289728000004214784000001258291200002388787200026214400001268776964893716509696
1110000000000017600000000007920000000000168960000000002112000000000017031168000000927252480000034603008000008758886400001441792000001395654656060397977678594621966336
1210000000000001920000000000095040000000000225280000000000316800000000000291962880000000185450496000000830472192000002627665920000057671680000008373927936007247757312028353495041250737128116224
13100000000000002080000000000001123200000000000292864000000000045760000000000004744396800000000344408064000000017993564160000006831931392000001874329600000003628702105600047110422528003685954355201315333734419745679767281664
141000000000000002240000000000000131040000000000003727360000000000064064000000000000738017280000000006027141120000000035987128320000000159411732480000005248122880000000127004573696000021984863846400025801680486400184146722816060397977600309548761569116160
Technical details
Node id
n-f5eaba3cd682fe6a · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-2e8ee919cea99b33

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:         10,          16;
  2:        100,         320,         144;
  3:       1000,        4800,        4320,        1024;
  4:      10000,       64000,       86400,       40960,        6400;
  5:     100000,      800000,     1440000,     1024000,      320000,       36864;
  6:    1000000,     9600000,    21600000,    20480000,     9600000,     2211840,      200704;
  7:   10000000,   112000000,   302400000,   358400000,   224000000,    77414400,    14049280,    1048576;
  8:  100000000,  1280000000,  4032000000,  5734400000,  4480000000,  2064384000,   561971200,   83886080,   5308416;
  9: 1000000000, 14400000000, 51840000000, 86016000000, 80640000000, 46448640000, 16859136000, 3774873600, 477757440, 26214400;
%F Formula
The row polynomials satisfy (25+(30*n-10)*x+4*n^2*x^2)*P(n,x) = (250+(300*n+300)*x+(40*n^2+200*n+120)*x^2+(16*n^2+32*n+16)*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[((250 + (300*n + 300)*x + (40*n^2 + 200*n + 120)*x^2 + (16*n^2 + 32*n + 16)*x^3)*P[n - 1, x])/(25 + (30*n - 10)*x + 4*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] = ((250 + (300*n + 300)*x + (40*n^2 + 200*n + 120)*x^2 + (16*n^2 + 32*n + 16)*x^3)*v[i - 1])/(25 + (30*n - 10)*x + 4*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.