{"id":"n-e253edaf0b910376","revision":0,"first_index":0,"fingerprint":"f6-049b3701fa7fb1d4","name":"one_plus_derivative of veronese_section_2_1 of Triangle whose (i,j)-th entry is binomial(i,j)*2^(i-j).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[[],["1"],["4"],["13","1"],["40","8"],["120","42","1"],["352","184","12"],["1008","728","87","1"],["2816","2688","496","16"],["7680","9408","2448","148","1"],["20480","31488","10944","1040","20"],["53504","101376","45408","6160","225","1"],["137216","315392","177408","32384","1880","24"],["346112","951808","658944","155584","13000","318","1"],["860160","2795520","2342912","695552","78624","3080","28"]],"degrees":[-1,0,0,1,1,2,2,3,3,4,4,5,5,6,6],"row_sums":["0","1","4","14","48","163","548","1824","6016","19685","63972","206674","664304","2125767","6775876"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["1"]],"recurrence":{"denominator":{"coeffs":[["-60","60","-8"],["70","-68","8"],["-12","8","0"],["2","0","0"]]},"terms":[{"coeff":{"coeffs":[["0","-24","52","-8"],["104","-98","-18","4"],["-12","9","4","0"],["4","1","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","80","-196","76","-8"],["0","-24","38","-8","0"],["0","8","-2","0","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":173672,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-77c65dbad13f675c"]},{"id":173673,"position":1,"claim":"one_signed_coefficients","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":0},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 0: the coefficients of P_n have one sign","region_exact":null,"edges":["e-77c65dbad13f675c"]},{"id":173674,"position":2,"claim":"hadamard_kernel","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":0},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-77c65dbad13f675c"]}],"region_exact":{},"incoming_edges":[{"id":"e-77c65dbad13f675c","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-e253edaf0b910376","premise_count":1,"theorem":"Laguerre's theorem on differential operators: if phi is a real polynomial with only real roots, then phi(D) maps every real-rooted polynomial to a real-rooted polynomial or zero (write phi(D) = C D^j prod (1 + c_i D); D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"e2fc8b3e27a672f7f70ebfd6a00ed60fc888bb6a83e2b6eccc9d28528dd74726","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-4e34a7abc01c81bb","role":"sequence","name":"veronese_section_2_1 of Triangle whose (i,j)-th entry is binomial(i,j)*2^(i-j).","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-b55a184cafc28c2e","kind":"fingerprint_related","node_a_id":"n-e253edaf0b910376","node_b_id":"n-317098d0c5ac3c9c","fingerprint":"f6-049b3701fa7fb1d4","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-049b3701fa7fb1d4"},"other_id":"n-317098d0c5ac3c9c","other":{"id":"n-317098d0c5ac3c9c","name":"one_plus_derivative of veronese_section_2_1 of Triangle read by rows: T(n,k) is the number of ternary words of length n having k levels (n >= 1, 0 <= k <= n-1). A level is a pair of identical consecutive letters.","first_index":0,"fingerprint":"f6-049b3701fa7fb1d4","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["3"],["12"]],"card_rows":[["3"],["12"],["39","3"],["120","24"]],"row_sums_preview":["0","3","12","42","144","489","1644","5472"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-51b4f5e7901c2b50","steps":["Veronese 2 (j=1)","D+1"],"edges":["e-7d561377a0a291f9","e-79b7fdaeb608cc94"],"text":"A120910 \u2192 Veronese 2 (j=1) \u2192 D+1"},"headline":"A120910 \u2192 Veronese 2 (j=1) \u2192 D+1","chain":null}}],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:    0;\n  1:    1;\n  2:    4;\n  3:   13,    1;\n  4:   40,    8;\n  5:  120,   42,    1;\n  6:  352,  184,   12;\n  7: 1008,  728,   87,   1;\n  8: 2816, 2688,  496,  16;\n  9: 7680, 9408, 2448, 148, 1;\n","recurrence_sentence":"The row polynomials satisfy (2*n^3-12*n^2+70*n-60+(8*n^2-68*n+60)*x+(8*n-8)*x^2)*P(n,x) = (4*n^3-12*n^2+104*n+(n^3+9*n^2-98*n-24)*x+(4*n^2-18*n+52)*x^2+(4*n-8)*x^3)*P(n-1,x) + ((8*n^2-24*n+80)*x-(2*n^2-38*n+196)*x^2-(8*n-76)*x^3-8*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((4*n^3 - 12*n^2 + 104*n + (n^3 + 9*n^2 - 98*n - 24)*x + (4*n^2 - 18*n + 52)*x^2 + (4*n - 8)*x^3)*P[n - 1, x] + ((8*n^2 - 24*n + 80)*x - (2*n^2 - 38*n + 196)*x^2 - (8*n - 76)*x^3 - 8*x^4)*D[P[n - 1, x], x])/(2*n^3 - 12*n^2 + 70*n - 60 + (8*n^2 - 68*n + 60)*x + (8*n - 8)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((4*n^3 - 12*n^2 + 104*n + (n^3 + 9*n^2 - 98*n - 24)*x + (4*n^2 - 18*n + 52)*x^2 + (4*n - 8)*x^3)*v[i - 1] + ((8*n^2 - 24*n + 80)*x - (2*n^2 - 38*n + 196)*x^2 - (8*n - 76)*x^3 - 8*x^4)*deriv(v[i - 1], x))/(2*n^3 - 12*n^2 + 70*n - 60 + (8*n^2 - 68*n + 60)*x + (8*n - 8)*x^2)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-af754f07daf65b5e","steps":["Veronese 2 (j=1)","D+1"],"edges":["e-868a2ac5701aa5c6","e-77c65dbad13f675c"],"text":"A038207 \u2192 Veronese 2 (j=1) \u2192 D+1"},"headline":"A038207 \u2192 Veronese 2 (j=1) \u2192 D+1","chain":null}