A101842: 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.

A101842 ↗ palindromicγ-positivenonnegative coefficients

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

Recurrence from A101842

\(P_{n}(x) = \left(1 + x + 2\,n x^{2}\right)\,P_{n-1}(x) + \left(x - x^{3}\right)\,P_{n-1}'(x)\)

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

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	x²⁹	P_n(1)
0	1	1																													2
1	1	3	3	1																											8
2	1	7	16	16	7	1																									48
3	1	15	61	115	115	61	15	1																							384
4	1	31	206	626	1056	1056	626	206	31	1																					3840
5	1	63	659	2989	7554	11774	11774	7554	2989	659	63	1																			46080
6	1	127	2052	13308	47349	105099	154624	154624	105099	47349	13308	2052	127	1																	645120
7	1	255	6297	56935	274677	824331	1660957	2337507	2337507	1660957	824331	274677	56935	6297	255	1															10321920
8	1	511	19162	237862	1518478	5960818	15747154	29428654	39984640	39984640	29428654	15747154	5960818	1518478	237862	19162	511	1													185794560
9	1	1023	58015	979297	8136844	40851316	137449588	328667020	578256262	763546234	763546234	578256262	328667020	137449588	40851316	8136844	979297	58015	1023	1											3715891200
10	1	2047	175088	3994640	42707787	269892789	1133187936	3379555488	7457267610	12486389862	16101629952	16101629952	12486389862	7457267610	3379555488	1133187936	269892789	42707787	3994640	175088	2047	1									81749606400
11	1	4095	527333	16196635	221035335	1737962681	8970948507	32757809253	88625970954	183014498550	294006065874	371644257582	371644257582	294006065874	183014498550	88625970954	32757809253	8970948507	1737962681	221035335	16196635	527333	4095	1							1961990553600
12	1	8191	1586118	65408058	1132974636	10988940756	68955308930	304054713470	991868620965	2475653713755	4835096758428	7498954635108	9319104528384	9319104528384	7498954635108	4835096758428	2475653713755	991868620965	304054713470	68955308930	10988940756	1132974636	65408058	1586118	8191	1					51011754393600
13	1	16383	4766571	263423125	5768348070	68571004506	518601545258	2732160772566	10609978480755	31525445314445	73515353233833	136908665513559	206008861638948	252270887452380	252270887452380	206008861638948	136908665513559	73515353233833	31525445314445	10609978480755	2732160772566	518601545258	68571004506	5768348070	263423125	4766571	16383	1			1428329123020800
14	1	32767	14316124	1058901412	29229094321	423779953231	3837222174992	23953020829424	109631201095037	383239807849091	1052393900501708	2315402800370996	4138300225028877	6065248599702771	7332475985461248	7332475985461248	6065248599702771	4138300225028877	2315402800370996	1052393900501708	383239807849091	109631201095037	23953020829424	3837222174992	423779953231	29229094321	1058901412	14316124	32767	1	42849873690624000

b-files: A101842, A101842 (reversed triangle) ?

Technical details

Node id: n-b3bdd177c3ec2fb9 · revision 0 · JSON
Definition source: oeis_catalog:A101842
Fingerprint: f6-0214d827b805d73a

Observed on the stored rows ?

Degrees: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29 (affine: slope 2, intercept 1)
Palindromic rows: yes
γ-positive (gamma_positive): yes
Nonnegative coefficients (nonn): yes
Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

Matches ?

A101842 reversed triangle 61 terms, 7 rows · tabf · nonn · catalog b-file?
A101842 triangle 61 terms, 7 rows · tabf · nonn · catalog b-file?

Row sums ? A000165, A052576

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0: 1,    1;
  1: 1,    3,     3,      1;
  2: 1,    7,    16,     16,       7,        1;
  3: 1,   15,    61,    115,     115,       61,        15,         1;
  4: 1,   31,   206,    626,    1056,     1056,       626,       206,        31,         1;
  5: 1,   63,   659,   2989,    7554,    11774,     11774,      7554,      2989,       659,        63,         1;
  6: 1,  127,  2052,  13308,   47349,   105099,    154624,    154624,    105099,     47349,     13308,      2052,       127,         1;
  7: 1,  255,  6297,  56935,  274677,   824331,   1660957,   2337507,   2337507,   1660957,    824331,    274677,     56935,      6297,      255,       1;
  8: 1,  511, 19162, 237862, 1518478,  5960818,  15747154,  29428654,  39984640,  39984640,  29428654,  15747154,   5960818,   1518478,   237862,   19162,    511,     1;
  9: 1, 1023, 58015, 979297, 8136844, 40851316, 137449588, 328667020, 578256262, 763546234, 763546234, 578256262, 328667020, 137449588, 40851316, 8136844, 979297, 58015, 1023, 1;

%F Formula

The row polynomials satisfy P(n,x) = (1+x+2*n*x^2)*P(n-1,x) + (x-x^3)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1 + x.

%t Mathematica

P[0, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[(1 + x + 2*n*x^2)*P[n - 1, x] + (x - x^3)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1 + x; for(n = 1, N, my(i = n + 1); v[i] = (1 + x + 2*n*x^2)*v[i - 1] + (x - x^3)*deriv(v[i - 1], 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

A101842 triangle, offset 1, from P₀

%e Example

Triangle begins:
   1: 1,    1;
   2: 1,    3,     3,      1;
   3: 1,    7,    16,     16,       7,        1;
   4: 1,   15,    61,    115,     115,       61,        15,         1;
   5: 1,   31,   206,    626,    1056,     1056,       626,       206,        31,         1;
   6: 1,   63,   659,   2989,    7554,    11774,     11774,      7554,      2989,       659,        63,         1;
   7: 1,  127,  2052,  13308,   47349,   105099,    154624,    154624,    105099,     47349,     13308,      2052,       127,         1;
   8: 1,  255,  6297,  56935,  274677,   824331,   1660957,   2337507,   2337507,   1660957,    824331,    274677,     56935,      6297,      255,       1;
   9: 1,  511, 19162, 237862, 1518478,  5960818,  15747154,  29428654,  39984640,  39984640,  29428654,  15747154,   5960818,   1518478,   237862,   19162,    511,     1;
  10: 1, 1023, 58015, 979297, 8136844, 40851316, 137449588, 328667020, 578256262, 763546234, 763546234, 578256262, 328667020, 137449588, 40851316, 8136844, 979297, 58015, 1023, 1;

%F Formula

The row polynomials satisfy P(n,x) = (1+x+(2*n-2)*x^2)*P(n-1,x) + (x-x^3)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1 + x.

%t Mathematica

P[1, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[(1 + x + (2*n - 2)*x^2)*P[n - 1, x] + (x - x^3)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1 + x; for(n = 2, N, my(i = n); v[i] = (1 + x + (2*n - 2)*x^2)*v[i - 1] + (x - x^3)*deriv(v[i - 1], x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

A101842 reversed triangle, offset 1, from P₀

%e Example

Triangle begins:
   1: 1,    1;
   2: 1,    3,     3,      1;
   3: 1,    7,    16,     16,       7,        1;
   4: 1,   15,    61,    115,     115,       61,        15,         1;
   5: 1,   31,   206,    626,    1056,     1056,       626,       206,        31,         1;
   6: 1,   63,   659,   2989,    7554,    11774,     11774,      7554,      2989,       659,        63,         1;
   7: 1,  127,  2052,  13308,   47349,   105099,    154624,    154624,    105099,     47349,     13308,      2052,       127,         1;
   8: 1,  255,  6297,  56935,  274677,   824331,   1660957,   2337507,   2337507,   1660957,    824331,    274677,     56935,      6297,      255,       1;
   9: 1,  511, 19162, 237862, 1518478,  5960818,  15747154,  29428654,  39984640,  39984640,  29428654,  15747154,   5960818,   1518478,   237862,   19162,    511,     1;
  10: 1, 1023, 58015, 979297, 8136844, 40851316, 137449588, 328667020, 578256262, 763546234, 763546234, 578256262, 328667020, 137449588, 40851316, 8136844, 979297, 58015, 1023, 1;

%F Formula

Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (1+x+(2*n-2)*x^2)*P(n-1,x) + (x-x^3)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1 + x.

%t Mathematica

P[1, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[(1 + x + (2*n - 2)*x^2)*P[n - 1, x] + (x - x^3)*D[P[n - 1, x], x]];
Table[Reverse[CoefficientList[P[n, x], x]], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1 + x; for(n = 2, N, my(i = n); v[i] = (1 + x + (2*n - 2)*x^2)*v[i - 1] + (x - x^3)*deriv(v[i - 1], x)); v};
row(n, p) = {my(c = Vecrev(p)); c = Vecrev(c); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Product of real-rooted factors (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Product of real-rooted factors (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Product of real-rooted factors (elementary)

Proof chains ? JSON

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

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