one_plus_derivative of Irregular triangle T(n, k) = [x^k]( p(n, x) ), where p(n, x) = (1/x)(1-x)^n(1+x)^(n+2)( Sum_{j >= 0} j^(n+1)x^j ), read by rows.

nonnegative coefficients

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

Recurrence

\(\left(72 - 144\,n + 48\,n^{2}\right)\,P_{n}(x) = \left(-246 + 623\,x - 946\,x^{2} + 48\,x^{3} - 648\,x^{4} + 189\,x^{5} + 540\,n - 440\,n x + 1800\,n x^{2} - 910\,n x^{3} + 2604\,n x^{4} - 270\,n x^{5} - 192\,n^{2} + 96\,n^{2} x - 1000\,n^{2} x^{2} + 808\,n^{2} x^{3} - 2912\,n^{2} x^{4} + 144\,n^{2} x^{5} + 192\,n^{3} x^{2} - 192\,n^{3} x^{3} + 1248\,n^{3} x^{4} - 32\,n^{3} x^{5} - 192\,n^{4} x^{4}\right)\,P_{n-2}(x) + \left(-18 + 183\,x + 333\,x^{2} - 792\,x^{3} + 98\,x^{4} - 147\,x^{5} + 63\,x^{6} + 36\,n - 156\,n x - 460\,n x^{2} + 770\,n x^{3} - 164\,n x^{4} + 1074\,n x^{5} - 48\,n x^{6} + 48\,n^{2} x + 120\,n^{2} x^{2} - 224\,n^{2} x^{3} + 48\,n^{2} x^{4} - 784\,n^{2} x^{5} + 16\,n^{2} x^{6} + 160\,n^{3} x^{5}\right)\,P_{n-2}'(x) + \left(-6\,x + 103\,x^{2} - 30\,x^{3} - 94\,x^{4} + 36\,x^{5} - 9\,x^{6} + 12\,n x - 144\,n x^{2} - 12\,n x^{3} + 48\,n x^{4} + 96\,n x^{6} + 48\,n^{2} x^{2} - 16\,n^{2} x^{4} - 32\,n^{2} x^{6}\right)\,P_{n-2}''(x) + \left(258 - 37\,x - 354\,x^{2} + 63\,x^{3} - 540\,n - 236\,n x + 912\,n x^{2} - 48\,n x^{3} + 192\,n^{2} + 96\,n^{2} x - 784\,n^{2} x^{2} + 16\,n^{2} x^{3} + 192\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(6 - 31\,x + 36\,x^{2} - 81\,x^{3} - 12\,n + 240\,n x^{3} - 80\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 3 + 4\,x + x^{2}\)

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	3	4	1																											8
2	7	26	28	10	1																									72
3	15	108	251	256	117	20	1																							768
4	31	382	1526	2874	2856	1506	386	38	1																					9600
5	63	1256	7773	23040	38222	37584	21906	7168	1155	72	1																			138240
6	127	3978	36072	155250	381627	582084	569416	356868	138405	30642	3312	138	1																	2257920
7	255	12340	158719	946032	3226515	6963012	10000083	9760384	6465325	2838396	783597	123408	9345	268	1															41287680
8	511	37814	676282	5416282	24607018	71175174	139486954	191475770	186729040	129199090	62362654	20252766	4148998	480338	26302	526	1													836075520
9	1023	115008	2823905	29780464	175398420	656513648	1677792268	3052070640	4044699674	3944164720	2826798342	1470607952	541090900	134640336	21049100	1834192	74367	1040	1											18579456000
10	2047	348130	11637836	159366354	1193919189	5644094616	18243800400	42316521816	72538361094	93457831884	91166593992	67265217900	37185757434	15107912856	4367280336	853185624	103853355	6936354	211916	2066	1									449622835200
11	4095	1050572	47543427	837121152	7868365649	46162687332	184674578181	531771537216	1140678411222	1862688756984	2345106065502	2288923102272	1730501934882	1005999121608	443098431786	144333025728	33492560379	5229115644	503156463	26138176	609213	4116	1							11771943321600
12	8191	3164046	193068318	4340408154	50668125756	364132872330	1774169918210	6219814562190	16306088062395	32791728942060	51425576337708	63509970467748	62031050571504	47880756678132	29038356868548	13678395016380	4912175833605	1307658429030	247511641430	31302686370	2409577236	98544546	1766298	8214	1					331576403558400
13	16383	9516760	780768997	22297373472	320816321994	2796302624688	16391884112646	68943483949280	217062144437405	526424745825000	1002762662214519	1519666943296320	1846439993254716	1804775893047456	1418204457020292	891757653165504	444588751479753	173192530023720	51589148500995	11387424874400	1779828132698	184274182320	11459152662	372677472	5159739	16408	1			9998303861145600
14	32767	28599482	3148137520	113782523122	2006161861087	21045355665132	147021385617800	733469849840524	2736199814932691	7876858997048030	17886691807918376	32531184792044406	47874175904846355	57361659215885256	56109557841462096	44782818463814856	29052362637104685	15206059093847382	6346553916439808	2076516140158270	519859083722957	96227181488012	12540322668200	1072022907372	54299927617	1415985362	15168040	32794	1	321374052679680000

Technical details

Node id: n-6dcdf88cf7bb3ab0 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-97627a1032bb0c6f

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:    3,      4,       1;
  2:    7,     26,      28,       10,         1;
  3:   15,    108,     251,      256,       117,        20,          1;
  4:   31,    382,    1526,     2874,      2856,      1506,        386,         38,          1;
  5:   63,   1256,    7773,    23040,     38222,     37584,      21906,       7168,       1155,         72,          1;
  6:  127,   3978,   36072,   155250,    381627,    582084,     569416,     356868,     138405,      30642,       3312,        138,         1;
  7:  255,  12340,  158719,   946032,   3226515,   6963012,   10000083,    9760384,    6465325,    2838396,     783597,     123408,      9345,       268,        1;
  8:  511,  37814,  676282,  5416282,  24607018,  71175174,  139486954,  191475770,  186729040,  129199090,   62362654,   20252766,   4148998,    480338,    26302,     526,     1;
  9: 1023, 115008, 2823905, 29780464, 175398420, 656513648, 1677792268, 3052070640, 4044699674, 3944164720, 2826798342, 1470607952, 541090900, 134640336, 21049100, 1834192, 74367, 1040, 1;

%F Formula

The row polynomials satisfy (48*n^2-144*n+72)*P(n,x) = -(192*n^2-540*n+246-(96*n^2-440*n+623)*x-(192*n^3-1000*n^2+1800*n-946)*x^2+(192*n^3-808*n^2+910*n-48)*x^3+(192*n^4-1248*n^3+2912*n^2-2604*n+648)*x^4+(32*n^3-144*n^2+270*n-189)*x^5)*P(n-2,x) + (36*n-18+(48*n^2-156*n+183)*x+(120*n^2-460*n+333)*x^2-(224*n^2-770*n+792)*x^3+(48*n^2-164*n+98)*x^4+(160*n^3-784*n^2+1074*n-147)*x^5+(16*n^2-48*n+63)*x^6)*d/dx P(n-2,x) + ((12*n-6)*x+(48*n^2-144*n+103)*x^2-(12*n+30)*x^3-(16*n^2-48*n+94)*x^4+36*x^5-(32*n^2-96*n+9)*x^6)*d^2/dx^2 P(n-2,x) + (192*n^2-540*n+258+(96*n^2-236*n-37)*x+(192*n^3-784*n^2+912*n-354)*x^2+(16*n^2-48*n+63)*x^3)*P(n-1,x) - (12*n-6+31*x-36*x^2+(80*n^2-240*n+81)*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 4*x + x^2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 3 + 4*x + x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(192*n^2 - 540*n + 246 - (96*n^2 - 440*n + 623)*x - (192*n^3 - 1000*n^2 + 1800*n - 946)*x^2 + (192*n^3 - 808*n^2 + 910*n - 48)*x^3 + (192*n^4 - 1248*n^3 + 2912*n^2 - 2604*n + 648)*x^4 + (32*n^3 - 144*n^2 + 270*n - 189)*x^5)*P[n - 2, x] + (36*n - 18 + (48*n^2 - 156*n + 183)*x + (120*n^2 - 460*n + 333)*x^2 - (224*n^2 - 770*n + 792)*x^3 + (48*n^2 - 164*n + 98)*x^4 + (160*n^3 - 784*n^2 + 1074*n - 147)*x^5 + (16*n^2 - 48*n + 63)*x^6)*D[P[n - 2, x], x] + ((12*n - 6)*x + (48*n^2 - 144*n + 103)*x^2 - (12*n + 30)*x^3 - (16*n^2 - 48*n + 94)*x^4 + 36*x^5 - (32*n^2 - 96*n + 9)*x^6)*D[P[n - 2, x], {x, 2}] + (192*n^2 - 540*n + 258 + (96*n^2 - 236*n - 37)*x + (192*n^3 - 784*n^2 + 912*n - 354)*x^2 + (16*n^2 - 48*n + 63)*x^3)*P[n - 1, x] - (12*n - 6 + 31*x - 36*x^2 + (80*n^2 - 240*n + 81)*x^3)*D[P[n - 1, x], x])/(48*n^2 - 144*n + 72)]];
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] = 3 + 4*x + x^2; for(n = 2, N, my(i = n + 1); v[i] = (-(192*n^2 - 540*n + 246 - (96*n^2 - 440*n + 623)*x - (192*n^3 - 1000*n^2 + 1800*n - 946)*x^2 + (192*n^3 - 808*n^2 + 910*n - 48)*x^3 + (192*n^4 - 1248*n^3 + 2912*n^2 - 2604*n + 648)*x^4 + (32*n^3 - 144*n^2 + 270*n - 189)*x^5)*v[i - 2] + (36*n - 18 + (48*n^2 - 156*n + 183)*x + (120*n^2 - 460*n + 333)*x^2 - (224*n^2 - 770*n + 792)*x^3 + (48*n^2 - 164*n + 98)*x^4 + (160*n^3 - 784*n^2 + 1074*n - 147)*x^5 + (16*n^2 - 48*n + 63)*x^6)*deriv(v[i - 2], x) + ((12*n - 6)*x + (48*n^2 - 144*n + 103)*x^2 - (12*n + 30)*x^3 - (16*n^2 - 48*n + 94)*x^4 + 36*x^5 - (32*n^2 - 96*n + 9)*x^6)*deriv(deriv(v[i - 2], x), x) + (192*n^2 - 540*n + 258 + (96*n^2 - 236*n - 37)*x + (192*n^3 - 784*n^2 + 912*n - 354)*x^2 + (16*n^2 - 48*n + 63)*x^3)*v[i - 1] - (12*n - 6 + 31*x - 36*x^2 + (80*n^2 - 240*n + 81)*x^3)*deriv(v[i - 1], x))/(48*n^2 - 144*n + 72)); 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 differential operators
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from Irregular triangle T(n, k) = [x^k]( p(n, x) ), where p(n, x) = (1/x)*(1-x)^n*(1+x)^(n+2)*( Sum_{j >= 0} j^(n+1)*x^j ), …?
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Irregular triangle T(n, k) = [x^k]( p(n, x) ), where p(n, x) = (1/x)*(1-x)^n*(1+x)^(n+2)*( Sum_{j >= 0} j^(n+1)*x^j ), …?
    
    Product of real-rooted factors (elementary) polytool ? step ›
    
    from row_factor parent of Triangle read by rows: T(n,k), for k=-n..n-1, is the scaled (by 2^n n!) probability that the sum o…?
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
    
    Parent-row provers of polytool polytool ? step › direct check
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from row_factor parent of Triangle read by rows: T(n,k), for k=-n..n-1, is the scaled (by 2^n n!) probability that the sum o…?
    
    ↑ Liu–Wang theorem (above)
    
    ↑ Parent-row provers of polytool (above)
  - ↑ Product of real-rooted factors (elementary) (above)

References

[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.