A272866 → basis A013609 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(25 - 50\,x - 120\,x^{2} - 40\,x^{3} + 16\,x^{4} + 150\,n x + 160\,n x^{2} - 40\,n x^{3} - 64\,n x^{4} + 100\,n^{2} x^{2} + 160\,n^{2} x^{3} + 64\,n^{2} x^{4}\right)\,P_{n}(x) = \left(125 + 750\,x + 1800\,x^{2} + 2200\,x^{3} + 1440\,x^{4} + 480\,x^{5} + 64\,x^{6} + 750\,n x + 3300\,n x^{2} + 5600\,n x^{3} + 4560\,n x^{4} + 1760\,n x^{5} + 256\,n x^{6} + 500\,n^{2} x^{2} + 1800\,n^{2} x^{3} + 2320\,n^{2} x^{4} + 1280\,n^{2} x^{5} + 256\,n^{2} x^{6}\right)\,P_{n-1}(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	x¹⁷	x¹⁸	x¹⁹	x²⁰	x²¹	x²²	x²³	x²⁴	x²⁵	x²⁶	x²⁷	x²⁸	P_n(1)
0	1																													1
1	5	40	36																											81
2	25	400	1260	1280	400																									3365
3	125	3000	16200	35200	36000	17280	3136																							110941
4	625	20000	153000	512000	910000	921600	533120	163840	20736																					3234921
5	3125	125000	1237500	5600000	14250000	22320000	22344000	14336000	5702400	1280000	123904																			87321929
6	15625	750000	9112500	52000000	172500000	367200000	527240000	522240000	357696000	166400000	50181120	8847360	692224																	2234874829
7	78125	4375000	63000000	434000000	1776250000	4788000000	8986600000	12108800000	11884320000	8512000000	4401689600	1599897600	387645440	56197120	3686400															55006539285
8	390625	25000000	416250000	3360000000	16362500000	53424000000	124028800000	211968000000	272095200000	264960000000	196016128000	109412352000	45306060800	13487308800	2727936000	335544320	18939904													1313944410449
9	1953125	140625000	2657812500	24600000000	138937500000	532980000000	1475586000000	3059712000000	4863304800000	6008800000000	5811949440000	4405985280000	2605704192000	1188569088000	409743360000	103179878400	17898209280	1911029760	94633984											30651755802049
10	9765625	781250000	16523437500	172500000000	1108593750000	4887000000000	15721650000000	38361600000000	72797940000000	109208000000000	130825974400000	125807616000000	97207718400000	60150988800000	29569536000000	11387535360000	3359465472000	732561408000	111194931200	10485760000	462422016									701438137156341
11	48828125	4296875000	100546875000	1168750000000	8421875000000	41951250000000	153817125000000	431904000000000	953459100000000	1684540000000000	2410536832000000	2815013376000000	2693426278400000	2113086976000000	1356030720000000	707631513600000	297273630720000	98975416320000	25503858688000	4902092800000	661263482880	55826186240	2218786816							15798466995562061
12	244140625	23437500000	601171875000	7700000000000	61488281250000	341550000000000	1406116250000000	4466880000000000	11250657000000000	22893200000000000	38132830560000000	52465397760000000	59971572544000000	57128988672000000	45381219840000000	30006575104000000	16441849036800000	7410017894400000	2715640463360000	795869184000000	181974623846400	31262664294400	3794125455360	289910292480	10485760000					351095509252974265
13	1220703125	126953125000	3541992187500	49562500000000	434433593750000	2662481250000000	12161524375000000	43128800000000000	122084820000000000	281281000000000000	534898020800000000	848004019200000000	1128637215680000000	1266818183680000000	1202098809600000000	964840128512000000	654113969766400000	373289140224000000	178292791705600000	70662225920000000	22954108649472000	6005501984768000	1233811878707200	191582384947200	21128806400000	1474381742080	48922361856			7713868453981163961
14	6103515625	683593750000	20610351562500	312812500000000	2985937500000000	20014312500000000	100452909375000000	393421600000000000	1236687952500000000	3183180000000000000	6807484684000000000	12225927168000000000	18583811355200000000	24040522060800000000	26563543776000000000	25119892439040000000	20338826912768000000	14084268097536000000	8318948820582400000	4172257689600000000	1765036222709760000	623955327057920000	181945788530688000	42980408819712000	8015314944000000	1135273941401600	114722938552320	7365868912640	225754218496	167815750478515393181

Technical details

Node id: n-71f55a5a0c64a1c9 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-91642c520b1b8422

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:       5,        40,         36;
  2:      25,       400,       1260,        1280,          400;
  3:     125,      3000,      16200,       35200,        36000,        17280,          3136;
  4:     625,     20000,     153000,      512000,       910000,       921600,        533120,        163840,         20736;
  5:    3125,    125000,    1237500,     5600000,     14250000,     22320000,      22344000,      14336000,       5702400,       1280000,        123904;
  6:   15625,    750000,    9112500,    52000000,    172500000,    367200000,     527240000,     522240000,     357696000,     166400000,      50181120,       8847360,        692224;
  7:   78125,   4375000,   63000000,   434000000,   1776250000,   4788000000,    8986600000,   12108800000,   11884320000,    8512000000,    4401689600,    1599897600,     387645440,      56197120,      3686400;
  8:  390625,  25000000,  416250000,  3360000000,  16362500000,  53424000000,  124028800000,  211968000000,  272095200000,  264960000000,  196016128000,  109412352000,   45306060800,   13487308800,   2727936000,    335544320,    18939904;
  9: 1953125, 140625000, 2657812500, 24600000000, 138937500000, 532980000000, 1475586000000, 3059712000000, 4863304800000, 6008800000000, 5811949440000, 4405985280000, 2605704192000, 1188569088000, 409743360000, 103179878400, 17898209280, 1911029760, 94633984;

%F Formula

The row polynomials satisfy (25+(150*n-50)*x+(100*n^2+160*n-120)*x^2+(160*n^2-40*n-40)*x^3+(64*n^2-64*n+16)*x^4)*P(n,x) = (125+(750*n+750)*x+(500*n^2+3300*n+1800)*x^2+(1800*n^2+5600*n+2200)*x^3+(2320*n^2+4560*n+1440)*x^4+(1280*n^2+1760*n+480)*x^5+(256*n^2+256*n+64)*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[((125 + (750*n + 750)*x + (500*n^2 + 3300*n + 1800)*x^2 + (1800*n^2 + 5600*n + 2200)*x^3 + (2320*n^2 + 4560*n + 1440)*x^4 + (1280*n^2 + 1760*n + 480)*x^5 + (256*n^2 + 256*n + 64)*x^6)*P[n - 1, x])/(25 + (150*n - 50)*x + (100*n^2 + 160*n - 120)*x^2 + (160*n^2 - 40*n - 40)*x^3 + (64*n^2 - 64*n + 16)*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] = ((125 + (750*n + 750)*x + (500*n^2 + 3300*n + 1800)*x^2 + (1800*n^2 + 5600*n + 2200)*x^3 + (2320*n^2 + 4560*n + 1440)*x^4 + (1280*n^2 + 1760*n + 480)*x^5 + (256*n^2 + 256*n + 64)*x^6)*v[i - 1])/(25 + (150*n - 50)*x + (100*n^2 + 160*n - 120)*x^2 + (160*n^2 - 40*n - 40)*x^3 + (64*n^2 - 64*n + 16)*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

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

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
- from basis_map_A013609 of Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and…?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.?
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.?
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    
    Parent-row provers of polytool polytool ? step › direct check
    
    ↑ Iterated real-rootedness preservers (above)
    
    ↑ Parent-row provers of polytool (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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.