A062145: Triangle read by rows: T(n, k) = [z^k] P(n, z) where P(n, z) = Sum_{k=0..n} binomial(n, k) * Pochhammer(n - k + c, k) * z^k / k! and c = 4.

OEIS matches ?

• A002054: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 1 ?
• A062145: triangle, 55 terms agree, 10 full rows row-aligned (tabl) OEIS nonn in polytool catalog · view at the OEIS · generated b-file 990 terms from n = 0 ?

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

Triangle begins:
   1: 1;
   2: 1,   4;
   3: 1,  10,   10;
   4: 1,  18,   45,    20;
   5: 1,  28,  126,   140,    35;
   6: 1,  40,  280,   560,   350,    56;
   7: 1,  54,  540,  1680,  1890,   756,    84;
   8: 1,  70,  945,  4200,  7350,  5292,  1470,   120;
   9: 1,  88, 1540,  9240, 23100, 25872, 12936,  2640,  165;
  10: 1, 108, 2376, 18480, 62370, 99792, 77616, 28512, 4455, 220;

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

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

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

Triangle begins:
  0: 1;
  1: 1,   4;
  2: 1,  10,   10;
  3: 1,  18,   45,    20;
  4: 1,  28,  126,   140,    35;
  5: 1,  40,  280,   560,   350,    56;
  6: 1,  54,  540,  1680,  1890,   756,    84;
  7: 1,  70,  945,  4200,  7350,  5292,  1470,   120;
  8: 1,  88, 1540,  9240, 23100, 25872, 12936,  2640,  165;
  9: 1, 108, 2376, 18480, 62370, 99792, 77616, 28512, 4455, 220;

The row polynomials satisfy -((1/2)*n^2+(3/2)*n)*P(n,x) = -((1/2)*n^2+(3/2)*n+((3/2)*n^2+(7/2)*n+3)*x)*P(n-1,x) - ((n+3/2)*x-(n+3/2)*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-((1/2)*n^2 + (3/2)*n + ((3/2)*n^2 + (7/2)*n + 3)*x)*P[n - 1, x] - ((n + 3/2)*x - (n + 3/2)*x^2)*D[P[n - 1, x], x])/(-(1/2)*n^2 - (3/2)*n)]];
Table[PadRight[CoefficientList[P[n, x], x], n + 1], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-((1/2)*n^2 + (3/2)*n + ((3/2)*n^2 + (7/2)*n + 3)*x)*v[i - 1] - ((n + 3/2)*x - (n + 3/2)*x^2)*deriv(v[i - 1], x))/(-(1/2)*n^2 - (3/2)*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)

Proved properties

• Real-rooted. Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0)
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-62efc507a1cd208a
• Coefficients of one sign. Every nonzero row, n >= 1: the coefficients of P_n have one sign
  All coefficients of each polynomial have the same sign (after x -> -x when alternating).
  derived from the cited steps by a stated theorem: e-62efc507a1cd208a
• Hadamard kernel (Malo). Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)
  Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.
  derived from the cited steps by a stated theorem: e-62efc507a1cd208a
• Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m >= 2
  The roots of consecutive polynomials alternate on the real line.
  by the cited steps: e-62efc507a1cd208a

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

• theorem_check liu_wang e-62efc507a1cd208a root: theorem check

  Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?

Used in 9 proof steps

• e-a6e517cc00955d8a obreschkoff_combination → obreschkoff_combination of Triangle read by rows: T(n, k) = [z^k] P(n, z) where P(n, z) =…
• e-a6e517cc00955d8a obreschkoff_combination → obreschkoff_combination of Triangle read by rows: T(n, k) = [z^k] P(n, z) where P(n, z) =…
• e-20496aa59e8ba6c2 preserver_image (mobius, shift 0) → mobius_x_over_x_plus_one of Triangle read by rows: T(n, k) = [z^k] P(n, z) where P(n, z) …
• e-2270413d40e48a63 preserver_image (reversal, shift 0) → reversal of Triangle read by rows: T(n, k) = [z^k] P(n, z) where P(n, z) = Sum_{k=0..n} b…
• e-36b6691ae4af4747 preserver_image (derivative_polynomial, shift 0) → derivative of Triangle read by rows: T(n, k) = [z^k] P(n, z) where P(n, z) = Sum_{k=0..n}…
• e-42aebc790a856cde preserver_image (euler_polynomial, shift 0) → theta_plus_one_squared of Triangle read by rows: T(n, k) = [z^k] P(n, z) where P(n, z) = …
• e-78be3a4437894f86 preserver_image (euler_polynomial, shift 0) → theta_plus_one of Triangle read by rows: T(n, k) = [z^k] P(n, z) where P(n, z) = Sum_{k=0…
• e-b5db2586a8a89c95 preserver_image (affine, shift 0) → shift_x_plus_one of Triangle read by rows: T(n, k) = [z^k] P(n, z) where P(n, z) = Sum_{k…
• e-df4a805affbb39de preserver_image (derivative_polynomial, shift 0) → one_plus_derivative of Triangle read by rows: T(n, k) = [z^k] P(n, z) where P(n, z) = Sum…

Related sequences ?

Same fingerprint (1)

Equal up to content, sign, reversal and a small index shift.

• A062145 → reverse

References

1. [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