A169803: Triangle read by rows: T(n,k) = binomial(n+1-k,k) (n >= 0, 0 <= k <= n).

OEIS matches ?

• A212804: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A202278: row sums, 10 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A000045: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A248740: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A185357: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A274163: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A077373: row sums, 10 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A132636: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

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: 1, 1;
  2: 1, 2;
  3: 1, 3,  1;
  4: 1, 4,  3;
  5: 1, 5,  6,  1;
  6: 1, 6, 10,  4;
  7: 1, 7, 15, 10,  1;
  8: 1, 8, 21, 20,  5;
  9: 1, 9, 28, 35, 15, 1;

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

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

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

Proved properties

• Real-rooted. Every row P_n, n >= 0, 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-2d7abb76d58e6037, e-4d4f59f3aacb50cf
• 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-2d7abb76d58e6037, e-4d4f59f3aacb50cf
• 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-2d7abb76d58e6037, e-4d4f59f3aacb50cf
• Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m >= 1
  The roots of consecutive polynomials alternate on the real line.
  by the cited steps: e-4d4f59f3aacb50cf

Proof chains

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

• definition_equality e-a99385591de0c3db root: theorem check

  Ore-algebra identity of two definitions polytool ?

• parent_provers paper_theorem e-4d4f59f3aacb50cf root: theorem check

  Parent-row provers of polytool polytool ?

• preserver_image reversal, shift 0 e-2d7abb76d58e6037

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): Triangle read by rows: T(n,k) = binomial(k,n-k).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • basis_map A027466 e-1781530cbfe1e9bf

      Geometric basis map (elementary) polytool ?

      • premise (sequence): basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k).
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 7]
        • preserver_image affine, shift 0 e-742e5241e9a4e627

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Triangle read by rows: T(n,k) = binomial(k,n-k).
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-4, 0]
            • theorem_check ma_wang_interval e-f811a014585c6352 root: theorem check

              Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ?

    • basis_map A027467 e-361937c2693b166b

      Geometric basis map (elementary) polytool ?

      • premise (sequence): basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k).
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 15]
        • preserver_image affine, shift 0 e-8f58f94d7421b40d

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Triangle read by rows: T(n,k) = binomial(k,n-k).
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-4, 0]
            • ↵ theorem_check ma_wang_interval e-f811a014585c6352 (expanded above)
    • basis_map A027465 e-5c2b8bc6b43c730e

      Geometric basis map (elementary) polytool ?

      • premise (sequence): basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k).
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 3]
        • preserver_image affine, shift 0 e-a2e7017ebd9e3749

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Triangle read by rows: T(n,k) = binomial(k,n-k).
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-4, 0]
            • ↵ theorem_check ma_wang_interval e-f811a014585c6352 (expanded above)
    • basis_map A007318 e-ba7f4c6c74dd05c0

      Geometric basis map (elementary) polytool ?

      • premise (sequence): basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k).
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 1]
        • preserver_image affine, shift 0 e-51281bad32c8f5cd

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Triangle read by rows: T(n,k) = binomial(k,n-k).
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-4, 0]
            • ↵ theorem_check ma_wang_interval e-f811a014585c6352 (expanded above)
    • preserver_image reversal, shift 1 e-fb809a8e7c094d2b

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Irregular triangle read by rows: T(n,k) = binomial(n-k, k), n >= 0, 0 <= k <= floor(n/2); or, coefficients of (one vers…
        uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
        • parent_provers paper_theorem e-0801036cdfff24d2 root: theorem check

          Parent-row provers of polytool polytool ?

        • parent_provers paper_theorem e-0d3631d0f359c9ce root: theorem check

          Parent-row provers of polytool polytool ?

        • gamma_polynomial scale 1, shift 0 e-a039cba7823b0b25

          Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ?

          • premise (sequence): delannoy
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
            • theorem_check liu_wang_derivative_lag e-15c233e4385971d1 root: theorem check

              Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ?

            • preserver_image reversal, shift 0 e-266d85ca03a298fa

              Affine, reversal and Möbius substitutions (elementary) polytool ?

              • premise (sequence): delannoy
                uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
                • ↵ theorem_check liu_wang_derivative_lag e-15c233e4385971d1 (expanded above)
                • basis_map A008288 e-e771e35dc9ee3fbd

                  Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ?

                  • premise (sequence): a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.
                    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in [0, 0]
                    no recorded property covers this requirement
                • basis_map A008288 e-33dba7e79fe9caa8

                  Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ?

                  • premise (sequence): a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.
                    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in [0, 0]
                    no recorded property covers this requirement
            • ↵ basis_map A008288 e-e771e35dc9ee3fbd (expanded above)
            • ↵ basis_map A008288 e-33dba7e79fe9caa8 (expanded above)
    • basis_map A030528 e-be49b310b519d57a

      Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ?

      • premise (sequence): a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in [0, 0]
        no recorded property covers this requirement
    • ↵ theorem_check ma_wang_interval e-f811a014585c6352 (expanded above)

Used in 12 proof steps

• e-3c621098a247eb31 basis_map (A008277) → basis_map_A008277 of Triangle read by rows: T(n,k) = binomial(n+1-k,k) (n >= 0, 0 <= k <=…
• e-43029c72e0d48967 basis_map (A013609) → basis_map_A013609 of Triangle read by rows: T(n,k) = binomial(n+1-k,k) (n >= 0, 0 <= k <=…
• e-62f41b682ce1c6f7 basis_map (A013610) → basis_map_A013610 of Triangle read by rows: T(n,k) = binomial(n+1-k,k) (n >= 0, 0 <= k <=…
• e-7c086361bc3430d6 basis_map (A013611) → basis_map_A013611 of Triangle read by rows: T(n,k) = binomial(n+1-k,k) (n >= 0, 0 <= k <=…
• e-eae4e58fb0906170 basis_map (A007318) → shift_x_plus_one of Triangle read by rows: T(n,k) = binomial(n+1-k,k) (n >= 0, 0 <= k <= …
• e-1b4d12a0914b0a40 preserver_image (affine, shift 0) → shift_x_plus_one of Triangle read by rows: T(n,k) = binomial(n+1-k,k) (n >= 0, 0 <= k <= …
• e-917662e9096bb187 preserver_image (derivative_polynomial, shift 0) → derivative of Triangle read by rows: T(n,k) = binomial(n+1-k,k) (n >= 0, 0 <= k <= n).
• e-c190684bc410f050 preserver_image (derivative_polynomial, shift 0) → one_plus_derivative of Triangle read by rows: T(n,k) = binomial(n+1-k,k) (n >= 0, 0 <= k …
• e-cd5540b5c0539e97 preserver_image (euler_polynomial, shift 0) → theta_plus_one_squared of Triangle read by rows: T(n,k) = binomial(n+1-k,k) (n >= 0, 0 <=…
• e-fdcd1c9a322f2427 preserver_image (euler_polynomial, shift 0) → theta_plus_one of Triangle read by rows: T(n,k) = binomial(n+1-k,k) (n >= 0, 0 <= k <= n).
• e-652f48e015a01bf3 veronese_section (r=2, j=1, shift 0) → veronese_section_2_1 of Triangle read by rows: T(n,k) = binomial(n+1-k,k) (n >= 0, 0 <= k…
• e-fc2a267a1cf18403 veronese_section (r=2, j=0, shift 0) → veronese_section_2_0 of Triangle read by rows: T(n,k) = binomial(n+1-k,k) (n >= 0, 0 <= k…

Related sequences ?

Same fingerprint (5)

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

• A011973: Irregular triangle read by rows: T(n,k) = binomial(n-k, k), n >= 0, 0 <= k <= floor(n/2); or, coefficients of …
• A052553: Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by upward antidiagonals.
• A026729: Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.
• A113214 → θ+1
• A030528: Triangle read by rows: T(n,k) = binomial(k,n-k).

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
2. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741