derivative of Triangle read by rows: T(n,k), for k=-n..n-1, is the scaled (by 2^n n!) probability that the sum of n uniform [-1, 1] variables is between k and k+1.

nonnegative coefficients

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

Recurrence

\(\left(1 - 4\,x + 4\,n x\right)\,P_{n}(x) = \left(-2 - 2\,x + 4\,x^{2} + 6\,x^{3} - 6\,x^{4} - 8\,n x - 10\,n x^{2} + 12\,n x^{3} + 34\,n x^{4} - 24\,n x^{5} - 8\,n^{2} x^{3} - 20\,n^{2} x^{4} + 40\,n^{2} x^{5} - 16\,n^{3} x^{5}\right)\,P_{n-2}(x) + \left(-x - x^{2} + 3\,x^{3} + x^{4} - 2\,x^{5} - 4\,n x^{2} - 6\,n x^{3} + 12\,n x^{4} + 6\,n x^{5} - 8\,n x^{6} - 8\,n^{2} x^{4} + 8\,n^{2} x^{6}\right)\,P_{n-2}'(x) + \left(3 - 6\,x - 9\,x^{2} + 12\,x^{3} + 12\,n x + 12\,n x^{2} - 28\,n x^{3} + 16\,n^{2} x^{3}\right)\,P_{n-1}(x) + \left(x - 4\,x^{2} - x^{3} + 4\,x^{4} + 4\,n x^{2} - 4\,n x^{4}\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 3 + 6\,x + 3\,x^{2}\)

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	6	3																											12
2	7	32	48	28	5																									120
3	15	122	345	460	305	90	7																							1344
4	31	412	1878	4224	5280	3756	1442	248	9																					17280
5	63	1318	8967	30216	58870	70644	52878	23912	5931	630	11																			253440
6	127	4104	39924	189396	525495	927744	1082368	840792	426141	133080	22572	1524	13																	4193280
7	255	12594	170805	1098708	4121655	9965742	16362549	18700056	14948613	8243310	3021447	683220	81861	3570	15															77414400
8	511	38324	713586	6073912	29804090	94482924	206000578	319877120	359861760	294286540	173218694	71529816	19740214	3330068	287430	8176	17													1579253760
9	1023	116030	2937891	32547376	204256580	824697528	2300669140	4626050096	6871916106	7635462340	6360818882	3944004240	1786844644	571918424	122052660	15668752	986255	18414	19											35300966400
10	2047	350176	11983920	170831148	1349463945	6799127616	23656888416	59658140880	112377508758	161016299520	177117929472	149836678344	96944478930	47313776832	16997819040	4318284624	726032379	71903520	3326672	40940	21									858370867200
11	4095	1054666	48589905	884141340	8689813405	53825691042	229304664771	709007767632	1647130486950	2940060658740	4088086833402	4459731090984	3822078856362	2562202979700	1329389564310	524124948048	152506124619	31283328258	4199671365	323932700	11073993	90090	23							22562891366400
12	8191	3172236	196224174	4531898544	54944703780	413731853580	2128382994290	7934948967720	22280883423795	48350967584280	82488500986188	111829254340608	121148358868992	104985364891512	72526451376420	39610459420080	16861766556405	5472984842460	1310150869670	219778815120	23792467356	1438977276	36480714	196584	25					637646929920000
13	16383	9533142	790269375	23073392280	342855022530	3111609271548	19125125407962	84879827846040	283729007830005	735153532338330	1505995320649149	2472106339667376	3279521536880940	3531792424333320	3090132924584220	2190538648216944	1249761004975161	567458015660010	201589591134345	54643215451320	10890632450418	1508562099132	132672005610	6322155000	119164275	425958	27			19282443160780800
14	32767	28632248	3176704236	116916377284	2118899766155	23023333049952	167671145805968	877049608760296	3449158270641819	10523939005017080	25469430804080956	49659602700346524	78848231796136023	102654663796457472	109987139781918720	97043977595244336	70351103825490909	41677250406677928	19995484109532452	7664796156981820	2302255222995777	526966458247328	88256110024816	10170718877544	730727358025	27531436712	386535348	917476	29	621323168514048000

Technical details

Node id: n-e679a7daf98af7e7 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-4649fccefba70494

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,      6,       3;
  2:    7,     32,      48,       28,         5;
  3:   15,    122,     345,      460,       305,        90,          7;
  4:   31,    412,    1878,     4224,      5280,      3756,       1442,        248,          9;
  5:   63,   1318,    8967,    30216,     58870,     70644,      52878,      23912,       5931,        630,         11;
  6:  127,   4104,   39924,   189396,    525495,    927744,    1082368,     840792,     426141,     133080,      22572,       1524,         13;
  7:  255,  12594,  170805,  1098708,   4121655,   9965742,   16362549,   18700056,   14948613,    8243310,    3021447,     683220,      81861,      3570,        15;
  8:  511,  38324,  713586,  6073912,  29804090,  94482924,  206000578,  319877120,  359861760,  294286540,  173218694,   71529816,   19740214,   3330068,    287430,     8176,     17;
  9: 1023, 116030, 2937891, 32547376, 204256580, 824697528, 2300669140, 4626050096, 6871916106, 7635462340, 6360818882, 3944004240, 1786844644, 571918424, 122052660, 15668752, 986255, 18414, 19;

%F Formula

The row polynomials satisfy (1+(4*n-4)*x)*P(n,x) = -(2+(8*n+2)*x+(10*n-4)*x^2+(8*n^2-12*n-6)*x^3+(20*n^2-34*n+6)*x^4+(16*n^3-40*n^2+24*n)*x^5)*P(n-2,x) - (x+(4*n+1)*x^2+(6*n-3)*x^3+(8*n^2-12*n-1)*x^4-(6*n-2)*x^5-(8*n^2-8*n)*x^6)*d/dx P(n-2,x) + (3+(12*n-6)*x+(12*n-9)*x^2+(16*n^2-28*n+12)*x^3)*P(n-1,x) + (x+(4*n-4)*x^2-x^3-(4*n-4)*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 6*x + 3*x^2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 3 + 6*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(2 + (8*n + 2)*x + (10*n - 4)*x^2 + (8*n^2 - 12*n - 6)*x^3 + (20*n^2 - 34*n + 6)*x^4 + (16*n^3 - 40*n^2 + 24*n)*x^5)*P[n - 2, x] - (x + (4*n + 1)*x^2 + (6*n - 3)*x^3 + (8*n^2 - 12*n - 1)*x^4 - (6*n - 2)*x^5 - (8*n^2 - 8*n)*x^6)*D[P[n - 2, x], x] + (3 + (12*n - 6)*x + (12*n - 9)*x^2 + (16*n^2 - 28*n + 12)*x^3)*P[n - 1, x] + (x + (4*n - 4)*x^2 - x^3 - (4*n - 4)*x^4)*D[P[n - 1, x], x])/(1 + (4*n - 4)*x)]];
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 + 6*x + 3*x^2; for(n = 2, N, my(i = n + 1); v[i] = (-(2 + (8*n + 2)*x + (10*n - 4)*x^2 + (8*n^2 - 12*n - 6)*x^3 + (20*n^2 - 34*n + 6)*x^4 + (16*n^3 - 40*n^2 + 24*n)*x^5)*v[i - 2] - (x + (4*n + 1)*x^2 + (6*n - 3)*x^3 + (8*n^2 - 12*n - 1)*x^4 - (6*n - 2)*x^5 - (8*n^2 - 8*n)*x^6)*deriv(v[i - 2], x) + (3 + (12*n - 6)*x + (12*n - 9)*x^2 + (16*n^2 - 28*n + 12)*x^3)*v[i - 1] + (x + (4*n - 4)*x^2 - x^3 - (4*n - 4)*x^4)*deriv(v[i - 1], x))/(1 + (4*n - 4)*x)); 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 Triangle read by rows: T(n,k), for k=-n..n-1, is the scaled (by 2^n n!) probability that the sum of n uniform [-1, 1] v…?
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n,k), for k=-n..n-1, is the scaled (by 2^n n!) probability that the sum of n uniform [-1, 1] v…?
    
    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.