A145596 → x ↦ x+1

OEIS matches ?

• A259476: reversed triangle, 66 terms agree, 11 full rows row-aligned (tabl) OEIS nonn · view at the OEIS · generated b-file 990 terms from n = 4 ?
• A065096: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

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:
  0:     1;
  1:     4,      2;
  2:    14,     14,      3;
  3:    48,     72,     32,      4;
  4:   165,    330,    225,     60,      5;
  5:   572,   1430,   1320,    550,    100,      6;
  6:  2002,   6006,   7007,   4004,   1155,    154,     7;
  7:  7072,  24752,  34944,  25480,  10192,   2184,   224,    8;
  8: 25194, 100776, 167076, 148512,  76440,  22932,  3822,  312,   9;
  9: 90440, 406980, 775200, 813960, 514080, 199920, 47040, 6300, 420, 10;

The row polynomials satisfy (n^2+4*n)*P(n,x) = (4*n^2+10*n+6+(3*n^2+5*n+2)*x)*P(n-1,x) - ((2*n+2)*x+(2*n+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[((4*n^2 + 10*n + 6 + (3*n^2 + 5*n + 2)*x)*P[n - 1, x] - ((2*n + 2)*x + (2*n + 2)*x^2)*D[P[n - 1, x], x])/(n^2 + 4*n)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((4*n^2 + 10*n + 6 + (3*n^2 + 5*n + 2)*x)*v[i - 1] - ((2*n + 2)*x + (2*n + 2)*x^2)*deriv(v[i - 1], x))/(n^2 + 4*n)); 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)

Triangle begins:
   4:  1;
   5:  2,   4;
   6:  3,  14,   14;
   7:  4,  32,   72,    48;
   8:  5,  60,  225,   330,    165;
   9:  6, 100,  550,  1320,   1430,    572;
  10:  7, 154, 1155,  4004,   7007,   6006,   2002;
  11:  8, 224, 2184, 10192,  25480,  34944,  24752,   7072;
  12:  9, 312, 3822, 22932,  76440, 148512, 167076, 100776,  25194;
  13: 10, 420, 6300, 47040, 199920, 514080, 813960, 775200, 406980, 90440;

Row n lists the coefficients of P(n,x) in descending powers of x, where (n^2-4*n)*P(n,x) = (4*n^2-22*n+30+(3*n^2-19*n+30)*x)*P(n-1,x) - ((2*n-6)*x+(2*n-6)*x^2)*d/dx P(n-1,x) for n >= 5, with P(4,x) = 1.

P[4, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^2 - 22*n + 30 + (3*n^2 - 19*n + 30)*x)*P[n - 1, x] - ((2*n - 6)*x + (2*n - 6)*x^2)*D[P[n - 1, x], x])/(n^2 - 4*n)]];
Table[Reverse[PadRight[CoefficientList[P[n, x], x], n - 3]], {n, 4, 13}] // Flatten

P(N) = {my(v = vector(N - 3)); v[1] = 1; for(n = 5, N, my(i = n - 3); v[i] = ((4*n^2 - 22*n + 30 + (3*n^2 - 19*n + 30)*x)*v[i - 1] - ((2*n - 6)*x + (2*n - 6)*x^2)*deriv(v[i - 1], x))/(n^2 - 4*n)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - 3 - #c)); c = Vecrev(c); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i + 3, v[i])))};
T(13)

Proved properties

• Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, -1]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-33d7b30ac5eafba9
• Coefficients of one sign. Every nonzero row, n >= 0: 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-33d7b30ac5eafba9
• Hadamard kernel (Malo). Rows P_n, n >= 0, 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-33d7b30ac5eafba9
• Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
  The roots of consecutive polynomials alternate on the real line.
  by the cited steps: e-33d7b30ac5eafba9

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

• preserver_image affine, shift 0 e-33d7b30ac5eafba9

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): basis-map coefficients of Triangular array of generalized Narayana numbers: T(n, k) = 2*binomial(n + 1, k + 1)*binomial…
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-c483eeee9c2638c9 root: theorem check

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

    • preserver_image reversal, shift 0 e-1d2c7b4a3cb3d72f

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): basis-map coefficients of Triangular array of generalized Narayana numbers: T(n, k) = 2*binomial(n + 1, k + 1)*binomial…
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • ↵ theorem_check liu_wang e-c483eeee9c2638c9 (expanded above)
        • basis_map A007318 e-fb63394f9047a486

          Geometric basis map (elementary) polytool ?

          • premise (sequence): basis-map coefficients of Triangular array of generalized Narayana numbers: T(n, k) = 2*binomial(n + 1, k + 1)*binomial…
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 1]
            • preserver_image affine, shift 0 e-3babd9179153850c

              Affine, reversal and Möbius substitutions (elementary) polytool ?

              • premise (sequence): basis-map coefficients of Triangular array of generalized Narayana numbers: T(n, k) = 2*binomial(n + 1, k + 1)*binomial…
                uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
                • ↵ theorem_check liu_wang e-c483eeee9c2638c9 (expanded above)
    • ↵ basis_map A007318 e-fb63394f9047a486 (expanded above)

Related sequences ?

Same fingerprint (2)

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

• A033282 → D
• A145596 → x ↦ x/(x+1)

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