{"id":"n-2024ae9472770903","revision":0,"first_index":0,"fingerprint":"f6-62f46345657f9dbc","name":"one_plus_derivative of basis_map_A013609 of Triangle read by rows: T(n,k) is the number of ternary words of length n having k runs (i.e., subwords of maximal length) of identical letters (1 <= k <= n).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["3"],["21","12"],["99","168","48"],["405","1188","1008","192"],["1539","6480","9504","5376","768"],["5589","30780","64800","63360","26880","3072"],["19683","134136","369360","518400","380160","129024","12288"],["67797","551124","1877904","3447360","3628800","2128896","602112","49152"],["229635","2169504","8817984","20030976","27578880","23224320","11354112","2752512","196608"],["767637","8266860","39051072","105815808","180278784","198567936","139345920","58392576","12386304","786432"],["2539107","30705480","165337200","520680960","1058158080","1442230272","1323786240","796262400","291962880","55050240","3145728"],["8325909","111720708","675520560","2424945600","5727490560","9311791104","10576355328","8320942080","4379443200","1427374080","242221056","12582912"],["27103491","399643632","2681296992","10808328960","29099347200","54983909376","74494328832","72523579392","49925652480","23357030400","6851395584","1056964608","50331648"],["87687765","1409381532","10390734432","46475814528","140508276480","302633210880","476527214592","553386442752","471403266048","288459325440","121456558080","32388415488","4580179968","201326592"],["282195171","4910514840","39462682896","193960376064","650661403392","1573692696576","2824576634880","3812217716736","3873705099264","2933175877632","1615372222464","618324295680","151145938944","19730006016","805306368"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["3","33","315","2793","23667","194481","1563051","12353145","96354531","743659329","5689858587","43218713097","326258912595","2449907834577","18312022966923"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["3"]],"recurrence":{"denominator":{"coeffs":[["-1","4"],["4","0"]]},"terms":[{"coeff":{"coeffs":[["9","24","16"],["12","16","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":183529,"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-763a999cfd495e02"]},{"id":183530,"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-763a999cfd495e02"]},{"id":183531,"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-763a999cfd495e02"]},{"id":183532,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-763a999cfd495e02"]}],"region_exact":{},"incoming_edges":[{"id":"e-763a999cfd495e02","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-2024ae9472770903","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":"d82722fb887997a241884a7099de78b0e94e74d12590d802957e982baf5ffd75","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-4a6fce4b90b6e934","role":"sequence","name":"basis_map_A013609 of Triangle read by rows: T(n,k) is the number of ternary words of length n having k runs (i.e., subwords of maximal length) of identical letters (1 <= k <= n).","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-6646bd445b48ebfa","kind":"fingerprint_related","node_a_id":"n-110aab533f3cf401","node_b_id":"n-2024ae9472770903","fingerprint":"f6-62f46345657f9dbc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-62f46345657f9dbc"},"other_id":"n-110aab533f3cf401","other":{"id":"n-110aab533f3cf401","name":"one_plus_derivative of basis_map_A013609 of Triangle of coefficients in expansion of (1+2*x)^n.","first_index":0,"fingerprint":"f6-62f46345657f9dbc","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["7","4"],["33","56","16"]],"card_rows":[["1"],["7","4"],["33","56","16"],["135","396","336","64"]],"row_sums_preview":["1","11","105","931","7889","64827","521017","4117715"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-71965e79e47a4526","steps":["basis A013609","D+1"],"edges":["e-95924f9cf72f1e84","e-58ab2b87ee1ed114"],"text":"A013609 \u2192 basis A013609 \u2192 D+1"},"headline":"A013609 \u2192 basis A013609 \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:      3;\n  1:     21,      12;\n  2:     99,     168,       48;\n  3:    405,    1188,     1008,       192;\n  4:   1539,    6480,     9504,      5376,       768;\n  5:   5589,   30780,    64800,     63360,     26880,      3072;\n  6:  19683,  134136,   369360,    518400,    380160,    129024,     12288;\n  7:  67797,  551124,  1877904,   3447360,   3628800,   2128896,    602112,    49152;\n  8: 229635, 2169504,  8817984,  20030976,  27578880,  23224320,  11354112,  2752512,   196608;\n  9: 767637, 8266860, 39051072, 105815808, 180278784, 198567936, 139345920, 58392576, 12386304, 786432;\n","recurrence_sentence":"The row polynomials satisfy (4*n-1+4*x)*P(n,x) = (12*n+9+(16*n+24)*x+16*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 3.","mathematica":"P[0, x_] := 3;\nP[n_, x_] := P[n, x] = Expand[Together[((12*n + 9 + (16*n + 24)*x + 16*x^2)*P[n - 1, x])/(4*n - 1 + 4*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 3; for(n = 1, N, my(i = n + 1); v[i] = ((12*n + 9 + (16*n + 24)*x + 16*x^2)*v[i - 1])/(4*n - 1 + 4*x)); 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-51b4f5e7901c2b50","steps":["basis A013611","D+1"],"edges":["e-1a05d30902084644","e-763a999cfd495e02"],"text":"A120910 \u2192 basis A013611 \u2192 D+1"},"headline":"A120910 \u2192 basis A013611 \u2192 D+1","chain":null}
