Real-rooted atlas

A272867 → basis A013609 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(9 - 18\,x - 36\,x^{2} - 12\,x^{3} + 4\,x^{4} + 54\,n x + 48\,n x^{2} - 12\,n x^{3} - 16\,n x^{4} + 36\,n^{2} x^{2} + 48\,n^{2} x^{3} + 16\,n^{2} x^{4}\right)\,P_{n}(x) = \left(54 + 324\,x + 756\,x^{2} + 864\,x^{3} + 504\,x^{4} + 144\,x^{5} + 16\,x^{6} + 324\,n x + 1368\,n x^{2} + 2160\,n x^{3} + 1584\,n x^{4} + 528\,n x^{5} + 64\,n x^{6} + 216\,n^{2} x^{2} + 720\,n^{2} x^{3} + 816\,n^{2} x^{4} + 384\,n^{2} x^{5} + 64\,n^{2} x^{6}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
16483690
236576172815364004276
32165184272165529650400207363136162184
412964147231104099532816416001492992752640196608207365453712
577763110403032640132710403196800046282752417715202359296080870401536000123904169984672
6466562239488268738561492992004743360009495429121257209856112538419268304384027648000071368704106168326922245027133760
72799361567641622338892815049359365943974400152374763522677125427233209450496295030149121881169920085240995842675441664552394752674365443686400143044209792

All 15 rows

Technical details
Node id
n-a9ac7753e25a7e6b · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-9c6b6869d5c10ef7

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:        6,        48,          36;
  2:       36,       576,        1728,         1536,          400;
  3:      216,      5184,       27216,        55296,        50400,         20736,          3136;
  4:     1296,     41472,      311040,       995328,      1641600,       1492992,        752640,         196608,          20736;
  5:     7776,    311040,     3032640,     13271040,     31968000,      46282752,      41771520,       23592960,        8087040,        1536000,         123904;
  6:    46656,   2239488,    26873856,    149299200,    474336000,     949542912,    1257209856,     1125384192,      683043840,      276480000,       71368704,       10616832,        692224;
  7:   279936,  15676416,   223388928,   1504935936,   5943974400,   15237476352,   26771254272,    33209450496,    29503014912,    18811699200,     8524099584,     2675441664,     552394752,      67436544,      3686400;
  8:  1679616, 107495424,  1773674496,  14046068736,  66363494400,  207681159168,  456107065344,   727210524672,   858404708352,   757510963200,   500580089856,   246140633088,   88615747584,   22658678784,   3892838400,    402653184,    18939904;
  9: 10077696, 725594112, 13604889600, 123834728448, 681252249600, 2519262756864, 6649057787904, 12988887072768, 19210829709312, 21807238348800, 19131772796928, 12988887072768, 6794804330496, 2709330591744, 807410073600, 173946175488, 25568870400, 2293235712, 94633984;
%F Formula
The row polynomials satisfy (9+(54*n-18)*x+(36*n^2+48*n-36)*x^2+(48*n^2-12*n-12)*x^3+(16*n^2-16*n+4)*x^4)*P(n,x) = (54+(324*n+324)*x+(216*n^2+1368*n+756)*x^2+(720*n^2+2160*n+864)*x^3+(816*n^2+1584*n+504)*x^4+(384*n^2+528*n+144)*x^5+(64*n^2+64*n+16)*x^6)*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[((54 + (324*n + 324)*x + (216*n^2 + 1368*n + 756)*x^2 + (720*n^2 + 2160*n + 864)*x^3 + (816*n^2 + 1584*n + 504)*x^4 + (384*n^2 + 528*n + 144)*x^5 + (64*n^2 + 64*n + 16)*x^6)*P[n - 1, x])/(9 + (54*n - 18)*x + (36*n^2 + 48*n - 36)*x^2 + (48*n^2 - 12*n - 12)*x^3 + (16*n^2 - 16*n + 4)*x^4)]];
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] = ((54 + (324*n + 324)*x + (216*n^2 + 1368*n + 756)*x^2 + (720*n^2 + 2160*n + 864)*x^3 + (816*n^2 + 1584*n + 504)*x^4 + (384*n^2 + 528*n + 144)*x^5 + (64*n^2 + 64*n + 16)*x^6)*v[i - 1])/(9 + (54*n - 18)*x + (36*n^2 + 48*n - 36)*x^2 + (48*n^2 - 12*n - 12)*x^3 + (16*n^2 - 16*n + 4)*x^4)); 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.