A253283: Triangle read by rows: coefficients of the partial fraction decomposition of [d^n/dx^n] (x/(1-x))^n/n!.

A253283 ↗ nonnegative coefficients

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

Recurrence from A253283

\(\left(1 - n\right)\,P_{n}(x) = \left(6 + 6\,x - 3\,n - 4\,n x\right)\,P_{n-1}(x) + \left(-9 - 14\,x + 3\,n + 4\,n x\right)\,P_{n-2}(x) + \left(4 - n\right)\,P_{n-3}(x)\)

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

\(P_{2}(x) = x\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	0	1														1
3	0	2	3													5
4	0	3	12	10												25
5	0	4	30	60	35											129
6	0	5	60	210	280	126										681
7	0	6	105	560	1260	1260	462									3653
8	0	7	168	1260	4200	6930	5544	1716								19825
9	0	8	252	2520	11550	27720	36036	24024	6435							108545
10	0	9	360	4620	27720	90090	168168	180180	102960	24310						598417
11	0	10	495	7920	60060	252252	630630	960960	875160	437580	92378					3317445
12	0	11	660	12870	120120	630630	2018016	4084080	5250960	4157010	1847560	352716				18474633
13	0	12	858	20020	225225	1441440	5717712	14702688	24942060	27713400	19399380	7759752	1352078			103274625
14	0	13	1092	30030	400400	3063060	14702688	46558512	99768240	145495350	142262120	89237148	32449872	5200300		579168825
15	0	14	1365	43680	680680	6126120	34918884	133024320	349188840	640179540	818007190	713897184	405623400	135207800	20058300	3256957317

b-files: A253283 ?

Technical details

Node id: n-b0d6ec97a8d172b8 · revision 0 · JSON
Definition source: oeis_catalog:A253283
Fingerprint: f6-2c1fea4a558d4e6c

Observed on the stored rows ?

Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept -1)
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 ?

A253283 triangle 55 terms, 10 rows · tabl · nonn · catalog b-file?

Row sums ? A002002

Submission text ?

Atlas rows n = 1, …, ascending powers of x

%e Example

Triangle begins:
   1: 1;
   2: 0, 1;
   3: 0, 2,   3;
   4: 0, 3,  12,   10;
   5: 0, 4,  30,   60,    35;
   6: 0, 5,  60,  210,   280,   126;
   7: 0, 6, 105,  560,  1260,  1260,    462;
   8: 0, 7, 168, 1260,  4200,  6930,   5544,   1716;
   9: 0, 8, 252, 2520, 11550, 27720,  36036,  24024,   6435;
  10: 0, 9, 360, 4620, 27720, 90090, 168168, 180180, 102960, 24310;

%F Formula

The row polynomials satisfy -(n-1)*P(n,x) = -(3*n-6+(4*n-6)*x)*P(n-1,x) + (3*n-9+(4*n-14)*x)*P(n-2,x) - (n-4)*P(n-3,x) for n >= 4, with P(1,x) = 1, P(2,x) = x, P(3,x) = 2*x + 3*x^2.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := x;
P[3, x_] := 2*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(3*n - 6 + (4*n - 6)*x)*P[n - 1, x] + (3*n - 9 + (4*n - 14)*x)*P[n - 2, x] - (n - 4)*P[n - 3, x])/(-n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = x; v[3] = 2*x + 3*x^2; for(n = 4, N, my(i = n); v[i] = (-(3*n - 6 + (4*n - 6)*x)*v[i - 1] + (3*n - 9 + (4*n - 14)*x)*v[i - 2] - (n - 4)*v[i - 3])/(-n + 1)); 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

A253283 triangle, offset 0, from P₁

%e Example

Triangle begins:
  0: 1;
  1: 0, 1;
  2: 0, 2,   3;
  3: 0, 3,  12,   10;
  4: 0, 4,  30,   60,    35;
  5: 0, 5,  60,  210,   280,   126;
  6: 0, 6, 105,  560,  1260,  1260,    462;
  7: 0, 7, 168, 1260,  4200,  6930,   5544,   1716;
  8: 0, 8, 252, 2520, 11550, 27720,  36036,  24024,   6435;
  9: 0, 9, 360, 4620, 27720, 90090, 168168, 180180, 102960, 24310;

%F Formula

The row polynomials satisfy -n*P(n,x) = -(3*n-3+(4*n-2)*x)*P(n-1,x) + (3*n-6+(4*n-10)*x)*P(n-2,x) - (n-3)*P(n-3,x) for n >= 3, with P(0,x) = 1, P(1,x) = x, P(2,x) = 2*x + 3*x^2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := x;
P[2, x_] := 2*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(3*n - 3 + (4*n - 2)*x)*P[n - 1, x] + (3*n - 6 + (4*n - 10)*x)*P[n - 2, x] - (n - 3)*P[n - 3, x])/(-n)]];
Table[PadRight[CoefficientList[P[n, x], x], n + 1], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = x; v[3] = 2*x + 3*x^2; for(n = 3, N, my(i = n + 1); v[i] = (-(3*n - 3 + (4*n - 2)*x)*v[i - 1] + (3*n - 6 + (4*n - 10)*x)*v[i - 2] - (n - 3)*v[i - 3])/(-n)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); 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 ≥ 1, is zero or real-rooted with all roots in (−∞, 0]
by Row-hypergeometric rows (multiplier sequences)
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Row-hypergeometric rows (multiplier sequences)
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Row-hypergeometric rows (multiplier sequences)

Proof chains ? JSON

Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check

Used in 12

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Related ?

Same fingerprint ?

References

[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.