Real-rooted atlas

A013622 → basis A013610 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(23 + 45\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1234568
2529207020254624
3121677141513972591125314432
4279841219006064273508383500410062521381376
56436343629642252463817504820512504715718751845281251453933568
614803588917378126108500170375221743575003253845937525464881250830376562598867482624
73404825447466313050352737054860758925178893751746230653125204992294062513369062656253736694531256722988818432

All 15 rows

Technical details
Node id
n-249359630d4f905d · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-e8b28356a6da96c8

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:             1;
  1:            23,             45;
  2:           529,           2070,            2025;
  3:         12167,          71415,          139725,            91125;
  4:        279841,        2190060,         6427350,          8383500,          4100625;
  5:       6436343,       62964225,       246381750,        482051250,        471571875,        184528125;
  6:     148035889,     1737812610,      8500170375,      22174357500,      32538459375,      25464881250,       8303765625;
  7:    3404825447,    46631305035,    273705486075,     892517889375,    1746230653125,    2049922940625,    1336906265625,     373669453125;
  8:   78310985281,  1225737160920,   8393634906300,   32844658329000,   80326610043750,  125728607025000,  122995376437500,   68755179375000,   16815125390625;
  9: 1801152661463, 31715949038805, 248211775086300, 1133140712350500, 3325521655811250, 6506455413543750, 8486680974187500, 7116161065312500, 3480730955859375, 756680642578125;
%F Formula
The row polynomials satisfy P(n,x) = (23+45*x)*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[(23 + 45*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; for(n = 1, N, my(i = n + 1); v[i] = (23 + 45*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. [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.