A008955 → basis A013610 → θ+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one sign Hadamard kernel interlacing (weak)

Recurrence

\(\left(1 - 2\,n + n^{2}\right)\,P_{n}(x) = \left(-4\,n^{2} - 12\,n^{2} x - 9\,n^{2} x^{2} + 8\,n^{3} + 36\,n^{3} x + 36\,n^{3} x^{2} - 8\,n^{4} - 42\,n^{4} x - 54\,n^{4} x^{2} + 4\,n^{5} + 24\,n^{5} x + 36\,n^{5} x^{2} - n^{6} - 6\,n^{6} x - 9\,n^{6} x^{2}\right)\,P_{n-2}(x) + \left(1 - 2\,n + 4\,n^{2} + 6\,n^{2} x - 4\,n^{3} - 12\,n^{3} x + 2\,n^{4} + 6\,n^{4} x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 1\)

\(P_{1}(x) = 2 + 6\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	2	6														8
2	10	78	108													196
3	100	1320	4239	3888												9547
4	1700	32040	167103	337392	233280											771515
5	44200	1088040	7949178	25482492	37695780	20995200										93254890
6	1635400	49804680	470382066	2087533836	4834880280	5662195488	2645395200									15751826950
7	81770000	2971041600	34501035240	196571576736	625328356365	1135982655792	1103336286192	444426393600								3543199115525
8	5315050000	224517384000	3098227271400	21609417509280	87823521580365	217914525932976	326176973499888	270991220668416	95996101017600							1023839819913925
9	435834100000	20993539788000	335891222722800	2775797871694560	13765389338033730	43478330019338466	88525279928989512	112805285306779008	81953905233671424	25918947274752000						369580255569869500
10	44019244100000	2381847978588000	43372106399602800	414712074130270560	2431228525026866730	9346851493645327866	24158468779596403812	41744858363066798208	46349128219638729024	29935782085973760000	8553252600668160000					162980080115368607000
11	5370347780200000	322543424604336000	6588313205092707600	71586972541300763520	484785483689888007660	2199359027726433147240	6905724798669557616315	15115186054001000334240	22702150176813328659120	22346313796409752759680	12996854604210837888000	3387088029864591360000				86215965170904167779375
12	778700428129000000	51408777049721520000	1164313553882052330000	14174979424622010288000	108950860307336173411500	570219853765170749520744	2109807045781208160574635	5601156964144715208748320	10637792197882418818013040	14137247585349811906531200	12503512233664486005159936	6616190879623909711872000	1585157197976628756480000			53885426298951667403449375
13	132379072781930000000	9529094332575464400000	237029679106262112060000	3196822464610009124040000	27505039462601348499975000	163223078551062355322083080	695952241284903885639208026	2174679166431541628243031480	5003184577064053136252039820	8395955027649897291590983200	10009940058072552721149639360	8040331397682858546963062784	3903419564289459297847296000	865495830095239301038080000		39283129494201291141175898750
14	26078677338040210000000	2032909373108916166800000	55099507985265195676620000	815605293947481293290920000	7768157285620822360664475000	51562502319370795500032726760	249073623419154841221872974002	896091701930469111913424995032	2424177630276083254924417168560	4922750397462211148794756372800	7402461903324246454267526887680	8004866835321698770154619307008	5890664754489004376091388305408	2642268051854210522295189504000	545262372960000759653990400000	33037820694432054050623227456250

Technical details

Node id: n-4c6e433b812c6520 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-c2cf76ef9990271d

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,              6;
  2:           10,             78,             108;
  3:          100,           1320,            4239,             3888;
  4:         1700,          32040,          167103,           337392,            233280;
  5:        44200,        1088040,         7949178,         25482492,          37695780,          20995200;
  6:      1635400,       49804680,       470382066,       2087533836,        4834880280,        5662195488,        2645395200;
  7:     81770000,     2971041600,     34501035240,     196571576736,      625328356365,     1135982655792,     1103336286192,       444426393600;
  8:   5315050000,   224517384000,   3098227271400,   21609417509280,    87823521580365,   217914525932976,   326176973499888,    270991220668416,    95996101017600;
  9: 435834100000, 20993539788000, 335891222722800, 2775797871694560, 13765389338033730, 43478330019338466, 88525279928989512, 112805285306779008, 81953905233671424, 25918947274752000;

%F Formula

The row polynomials satisfy (n^2-2*n+1)*P(n,x) = -(n^6-4*n^5+8*n^4-8*n^3+4*n^2+(6*n^6-24*n^5+42*n^4-36*n^3+12*n^2)*x+(9*n^6-36*n^5+54*n^4-36*n^3+9*n^2)*x^2)*P(n-2,x) + (2*n^4-4*n^3+4*n^2-2*n+1+(6*n^4-12*n^3+6*n^2)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 6*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + 6*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^6 - 4*n^5 + 8*n^4 - 8*n^3 + 4*n^2 + (6*n^6 - 24*n^5 + 42*n^4 - 36*n^3 + 12*n^2)*x + (9*n^6 - 36*n^5 + 54*n^4 - 36*n^3 + 9*n^2)*x^2)*P[n - 2, x] + (2*n^4 - 4*n^3 + 4*n^2 - 2*n + 1 + (6*n^4 - 12*n^3 + 6*n^2)*x)*P[n - 1, x])/(n^2 - 2*n + 1)]];
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 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^6 - 4*n^5 + 8*n^4 - 8*n^3 + 4*n^2 + (6*n^6 - 24*n^5 + 42*n^4 - 36*n^3 + 12*n^2)*x + (9*n^6 - 36*n^5 + 54*n^4 - 36*n^3 + 9*n^2)*x^2)*v[i - 2] + (2*n^4 - 4*n^3 + 4*n^2 - 2*n + 1 + (6*n^4 - 12*n^3 + 6*n^2)*x)*v[i - 1])/(n^2 - 2*n + 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

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Laguerre’s theorem on multiplier sequences
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on multiplier sequences
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
- from basis_map_A013610 of Triangle of central factorial numbers |t(2n,2n-2k)| read by rows.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle of central factorial numbers |t(2n,2n-2k)| read by rows.?
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
    
    Parent-row provers of polytool polytool ? step › direct check
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle of central factorial numbers |t(2n,2n-2k)| read by rows.?
    
    ↑ Liu–Wang theorem (above)
    
    ↑ Parent-row provers of polytool (above)
    
    ↑ Iterated real-rootedness preservers (above)

References

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