A272866 → basis A013611 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(5 - 16\,x^{2} + 20\,n x + 32\,n x^{2}\right)\,P_{n}(x) = \left(25 + 200\,x + 560\,x^{2} + 640\,x^{3} + 256\,x^{4} + 100\,n x + 560\,n x^{2} + 960\,n x^{3} + 512\,n x^{4}\right)\,P_{n-1}(x)\)

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

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	48																											93
2	25	400	1680	2560	1280																									5945
3	125	3000	21600	70400	115200	92160	28672																							331157
4	625	20000	204000	1024000	2912000	4915200	4874240	2621440	589824																					17161329
5	3125	125000	1650000	11200000	45600000	119040000	204288000	229376000	162201600	65536000	11534336																			850554061
6	15625	750000	12150000	104000000	552000000	1958400000	4820480000	8355840000	10174464000	8519680000	4671406080	1509949440	218103808																	40897238953
7	78125	4375000	84000000	868000000	5684000000	25536000000	82163200000	193740800000	338042880000	435814400000	409757286400	273049190400	122138132480	32883343360	4026531840															1923792217605
8	390625	25000000	555000000	6720000000	52360000000	284928000000	1133977600000	3391488000000	7739596800000	13565952000000	18247319552000	18673041408000	14274894233600	7892002406400	2979633561600	687194767360	73014444032													89002703163617
9	1953125	140625000	3543750000	49200000000	444600000000	2842560000000	13491072000000	48955392000000	138334003200000	307650560000000	541039656960000	751954821120000	820997259264000	695482712064000	447549014016000	211312390963200	68998649610240	13915694039040	1305670057984											4064326941622589
10	9765625	781250000	22031250000	345000000000	3547500000000	26064000000000	143740800000000	613785600000000	2070696960000000	5591449600000000	12178708889600000	21471166464000000	30627908812800000	35196921446400000	32297818521600000	23321672417280000	12950937010176000	5334349381632000	1534162318131200	274877906944000	23089744183296									183661265195012121
11	48828125	4296875000	134062500000	2337500000000	26950000000000	223740000000000	1406328000000000	6910464000000000	27120614400000000	86248448000000000	224399065088000000	480428949504000000	848636464332800000	1236457750528000000	1481147154432000000	1449229339852800000	1146007333109760000	720716987105280000	351878080626688000	128505421496320000	33018334182113280	5321636278435840	404620279021568							8228090157123421813
12	244140625	23437500000	801562500000	15400000000000	196762500000000	1821600000000000	12855920000000000	71470080000000000	320018688000000000	1172131840000000000	3549819863040000000	8954094551040000000	18895658549248000000	33428619657216000000	49568393723904000000	61453465812992000000	63384295204454400000	53958103636377600000	37467834401423360000	20863233137049600000	9086391579731558400	2980116315924070400	691900677126881280	101330991615836160	7036874417766400					365968806091305213265
13	1220703125	126953125000	4722656250000	99125000000000	1390187500000000	14199900000000000	111191080000000000	690060800000000000	3472634880000000000	14401587200000000000	49794143027200000000	144726019276800000000	355607541186560000000	741269611479040000000	1313012459765760000000	1975992583192576000000	2521647830741811200000	2718208565968896000000	2459915030993305600000	1852367855157248000000	1146148925296607232000	572475022652735488000	224999221656918425600	66962896959465062400	14179301951799296000	1902770842564034560	121597189939003392			16178022545362555420077
14	6103515625	683593750000	27480468750000	625625000000000	9555000000000000	106743000000000000	918426600000000000	6294745600000000000	35176901760000000000	162978816000000000000	633714937856000000000	2086558236672000000000	5855330100838400000000	14067139765862400000000	29014473415065600000000	51445539715153920000000	78407374150303744000000	102558510448902144000000	114776974716986982400000	109373231978250240000000	88132124869719490560000	59478598275007774720000	33179823852931252224000	15022741719512776704000	5378986804940636160000	1465133548774306611200	285145410406962954240	35308221078584688640	2089670227099910144	711404285680245480947849

Technical details

Node id: n-b552d7a4ad6358e9 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-6e3a14f88852ba15

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,         48;
  2:      25,       400,       1680,        2560,         1280;
  3:     125,      3000,      21600,       70400,       115200,         92160,          28672;
  4:     625,     20000,     204000,     1024000,      2912000,       4915200,        4874240,        2621440,          589824;
  5:    3125,    125000,    1650000,    11200000,     45600000,     119040000,      204288000,      229376000,       162201600,        65536000,        11534336;
  6:   15625,    750000,   12150000,   104000000,    552000000,    1958400000,     4820480000,     8355840000,     10174464000,      8519680000,      4671406080,      1509949440,       218103808;
  7:   78125,   4375000,   84000000,   868000000,   5684000000,   25536000000,    82163200000,   193740800000,    338042880000,    435814400000,    409757286400,    273049190400,    122138132480,     32883343360,      4026531840;
  8:  390625,  25000000,  555000000,  6720000000,  52360000000,  284928000000,  1133977600000,  3391488000000,   7739596800000,  13565952000000,  18247319552000,  18673041408000,  14274894233600,   7892002406400,   2979633561600,    687194767360,    73014444032;
  9: 1953125, 140625000, 3543750000, 49200000000, 444600000000, 2842560000000, 13491072000000, 48955392000000, 138334003200000, 307650560000000, 541039656960000, 751954821120000, 820997259264000, 695482712064000, 447549014016000, 211312390963200, 68998649610240, 13915694039040, 1305670057984;

%F Formula

The row polynomials satisfy (5+20*n*x+(32*n-16)*x^2)*P(n,x) = (25+(100*n+200)*x+(560*n+560)*x^2+(960*n+640)*x^3+(512*n+256)*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[((25 + (100*n + 200)*x + (560*n + 560)*x^2 + (960*n + 640)*x^3 + (512*n + 256)*x^4)*P[n - 1, x])/(5 + 20*n*x + (32*n - 16)*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] = ((25 + (100*n + 200)*x + (560*n + 560)*x^2 + (960*n + 640)*x^3 + (512*n + 256)*x^4)*v[i - 1])/(5 + 20*n*x + (32*n - 16)*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

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_A013611 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.