Real-rooted atlas

A272866 → basis A013611 → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernel

Recurrence

\(P_{n}(x) = \left(41 + 52\,x + 16\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1415216109
2168142644016166425611881
3689212622364132803452801612803993640961295029
428257611433556831683488398428163117952015548416482508885196865536141158161
51158562017346978602089684720351046592038571225602896305152150521856053460992012419072017039360104857615386239549
647501042413614713471212573506145626434787392037412120832037548633907227403578982414653125427256975032320157102899202916089856327155712167772161677100110841
71947542738811729037943724711079019257617954840181824310968199682563907880871731236746696347648262654239211521434017418444859513236357121848095277056416414695424643574005766106906624268435456182803912081669

All 15 rows

Technical details
Node id
n-b58f73cba181d1b6 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-10850f2bbd2ef26a

Observed on the stored rows ?

  • Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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:              41,               52,                16;
  2:            1681,             4264,              4016,              1664,                256;
  3:           68921,           262236,            413280,            345280,             161280,              39936,               4096;
  4:         2825761,         14335568,          31683488,          39842816,           31179520,           15548416,            4825088,             851968,              65536;
  5:       115856201,        734697860,        2089684720,        3510465920,         3857122560,         2896305152,         1505218560,          534609920,          124190720,           17039360,            1048576;
  6:      4750104241,      36147134712,      125735061456,      264347873920,       374121208320,       375486339072,       274035789824,       146531254272,        56975032320,        15710289920,         2916089856,         327155712,          16777216;
  7:    194754273881,    1729037943724,     7110790192576,    17954840181824,     31096819968256,     39078808717312,     36746696347648,     26265423921152,     14340174184448,      5951323635712,      1848095277056,      416414695424,       64357400576,       6106906624,        268435456;
  8:   7984925229121,   81017777934496,   384568439351360,  1133574144568320,   2322393951234560,   3506543238668288,   4036261723045888,   3612971530321920,   2541696327024640,   1409940109393920,    614683522367488,   208395135090688,    53861742018560,   10259603128320,    1358283407360,    111669149696,     4294967296;
  9: 327381934393961, 3736945007228628, 20107989269665488, 67770383220523776, 160317102547791360, 282669944562690048, 384985282275385344, 414122134160277504, 356664256553484288, 247783297975582720, 139186051337945088, 63066785452130304, 22879814805356544, 6555776474677248, 1450976854671360, 239362822373376, 27715423961088, 2010044694528, 68719476736;
%F Formula
The row polynomials satisfy P(n,x) = (41+52*x+16*x^2)*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[(41 + 52*x + 16*x^2)*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] = (41 + 52*x + 16*x^2)*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

References

  1. [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
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.