Real-rooted atlas

A034870 → Veronese 2 → basis A013609

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-4\,x^{2}\right)\,P_{n-2}(x) + \left(4 + 4\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1224
2816428
33296728208
4128512640256161552
55122560448032008003211584
620481228827648286721344023046486464
7819257344157696215040150528501766272128645376
83276826214485196814417921351680688128172032163842564817152
913107211796484423680894566410543104729907228385285529604147251235955712
10524288524288022282240524288007454720065601536351436801081344016896001024001024268377088
1120971522306867210957619229412556849020928052481228836080844815463219238658048495616024780820482003193856
1283886081006632965284823041593835520304821043238503710723244294144179935641663258624013120307214057472589824409614952042496
13335544324362076162508193792839699660818129879040264178237442641782374418075353088828453683224366284804264099843876454413844488192111603564544
1413421772818790481921174405120043218108416103993573376171815469056199149748224162571223040924623831043568723558489218088961335885824104366080321126416384833020346368
Technical details
Node id
n-55515e5d92a2ea3a · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-d44a70ca53ce7ac3

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:      2,       2;
  2:      8,      16,       4;
  3:     32,      96,      72,       8;
  4:    128,     512,     640,     256,       16;
  5:    512,    2560,    4480,    3200,      800,      32;
  6:   2048,   12288,   27648,   28672,    13440,    2304,      64;
  7:   8192,   57344,  157696,  215040,   150528,   50176,    6272,    128;
  8:  32768,  262144,  851968, 1441792,  1351680,  688128,  172032,  16384,   256;
  9: 131072, 1179648, 4423680, 8945664, 10543104, 7299072, 2838528, 552960, 41472, 512;
%F Formula
The row polynomials satisfy P(n,x) = -4*x^2*P(n-2,x) + (4+4*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + 2*x;
P[n_, x_] := P[n, x] = Expand[-4*x^2*P[n - 2, x] + (4 + 4*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = -4*x^2*v[i - 2] + (4 + 4*x)*v[i - 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.