Real-rooted atlas

A034870 → basis A013609 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(1 - 4\,x + x^{2} + 6\,n x - 4\,n x^{2} + 4\,n^{2} x^{2}\right)\,P_{n}(x) = \left(4 + 16\,x + 24\,x^{2} + 16\,x^{3} + 4\,x^{4} + 24\,n x + 64\,n x^{2} + 56\,n x^{3} + 16\,n x^{4} + 16\,n^{2} x^{2} + 32\,n^{2} x^{3} + 16\,n^{2} x^{4}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
14323672
21625686410244002560
36415368640204802400013824313671680
4256819264512229376448000516096351232131072207361769472
51024409604147201966080537600092897281053696078643203732480102400012390440370176
640961966082433024144179205068800011678515218545049620761804816422912090112000327106567077888692224872415232
716384917504134184969542041641000960011808276482410856448359871283239852933123280076800198444646485878374425196953644957696368640018119393280

All 15 rows

Technical details
Node id
n-79eab5fdeb711ab4 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-561a7d876fc231a6

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:      4,       32,        36;
  2:     16,      256,       864,       1024,         400;
  3:     64,     1536,      8640,      20480,       24000,       13824,         3136;
  4:    256,     8192,     64512,     229376,      448000,      516096,       351232,       131072,        20736;
  5:   1024,    40960,    414720,    1966080,     5376000,     9289728,     10536960,      7864320,      3732480,       1024000,        123904;
  6:   4096,   196608,   2433024,   14417920,    50688000,   116785152,    185450496,    207618048,    164229120,      90112000,      32710656,       7077888,       692224;
  7:  16384,   917504,  13418496,   95420416,   410009600,  1180827648,   2410856448,   3598712832,   3985293312,    3280076800,    1984446464,     858783744,    251969536,     44957696,      3686400;
  8:  65536,  4194304,  70778880,  587202560,  2981888000, 10305404928,  25715802112,  47982837760,  68319313920,   74973184000,   63502286848,   41221619712,  20157562880,   7193231360,   1769472000,   268435456,    18939904;
  9: 262144, 18874368, 360972288, 3422552064, 20054016000, 80857792512, 238455619584, 533918121984, 929142669312, 1274544128000, 1387978555392, 1201315774464, 822428565504, 440225759232, 180486144000, 54760833024, 11591221248, 1528823808, 94633984;
%F Formula
The row polynomials satisfy (1+(6*n-4)*x+(4*n^2-4*n+1)*x^2)*P(n,x) = (4+(24*n+16)*x+(16*n^2+64*n+24)*x^2+(32*n^2+56*n+16)*x^3+(16*n^2+16*n+4)*x^4)*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[((4 + (24*n + 16)*x + (16*n^2 + 64*n + 24)*x^2 + (32*n^2 + 56*n + 16)*x^3 + (16*n^2 + 16*n + 4)*x^4)*P[n - 1, x])/(1 + (6*n - 4)*x + (4*n^2 - 4*n + 1)*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] = ((4 + (24*n + 16)*x + (16*n^2 + 64*n + 24)*x^2 + (32*n^2 + 56*n + 16)*x^3 + (16*n^2 + 16*n + 4)*x^4)*v[i - 1])/(1 + (6*n - 4)*x + (4*n^2 - 4*n + 1)*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. [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.