{"id":"n-a7af5531257fc90d","revision":0,"first_index":0,"fingerprint":"f6-d31e0778cbd986bd","name":"one_plus_derivative of Triangle, read by rows of 2*n+1 terms, where row n lists the coefficients in (1+3*x+2*x^2)^n.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":2,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["4","7","2"],["7","32","49","28","4"],["10","75","222","327","246","84","8"],["13","136","602","1464","2121","1848","920","224","16"],["16","215","1270","4330","9400","13503","12890","7960","2960","560","32"],["19","312","2307","10140","29445","59352","84841","86196","61260","29280","8688","1344","64"],["22","427","3794","20433","74424","193641","370426","527887","561134","440412","249144","97104","23968","3136","128"],["25","560","5812","37072","162526","518784","1245748","2292176","3259921","3586288","3030832","1936704","909664","299264","63232","7168","256"],["28","711","8442","62244","319176","1207962","3494148","7891380","14090508","20011383","22616370","20241072","14187936","7642656","3068352","873216","161280","16128","512"],["31","880","11765","98460","578250","2532096","8573370","22970040","49422675","86163120","122246433","141194700","132281940","99727680","59696160","27787392","9740160","2442240","400640","35840","1024"],["34","1067","15862","148555","983290","4891194","18980412","58869822","148324110","306838455","524627862","743781591","874572006","850579620","680160360","442805088","231236544","94748544","29469440","6603520","974336","78848","2048"],["37","1272","20814","215688","1588719","8850072","38729372","136526544","394393923","944782344","1891929006","3182938056","4510043097","5383157544","5401036872","4534304544","3161570544","1810752768","838466816","306797568","85755648","17369088","2328576","172032","4096"],["40","1495","26702","303342","2461056","15178449","73957598","292098092","951948712","2592613881","5953896234","11600762238","19252648896","27268265391","32964226698","33956167752","29697720624","21925270224","13549394144","6926196992","2880537088","952064256","241632768","44648448","5484544","372736","8192"],["43","1736","33607","415324","3680131","24895416","133675451","584774476","2122725605","6479168696","16790290517","37199841588","70807743825","116151584808","164455126233","200996914020","211770414492","191790327456","148622089616","97896390592","54313907648","25067254784","9460994816","2851150848","662631424","112566272","12759040","802816","16384"]],"degrees":[0,2,4,6,8,10,12,14,16,18,20,22,24,26,28],"row_sums":["1","13","120","972","7344","53136","373248","2566080","17356032","115893504","765904896","5018692608","32651735040","211147886592","1358312177664"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["-2","-1","2"],["3","4","0"]]},"terms":[{"coeff":{"coeffs":[["1","6","13","12","4"],["3","13","18","8","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":179669,"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-97f1ae3db6b837a6"]},{"id":179670,"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-97f1ae3db6b837a6"]},{"id":179671,"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-97f1ae3db6b837a6"]}],"region_exact":{},"incoming_edges":[{"id":"e-97f1ae3db6b837a6","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-a7af5531257fc90d","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":"47629a3b0ed9f55b7ca3d9524277e3d86c17549a6283b4be3c690c334759c812","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-eed4f744ce063213","role":"sequence","name":"Triangle, read by rows of 2*n+1 terms, where row n lists the coefficients in (1+3*x+2*x^2)^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":[],"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:  1;\n  1:  4,   7,    2;\n  2:  7,  32,   49,    28,      4;\n  3: 10,  75,  222,   327,    246,      84,       8;\n  4: 13, 136,  602,  1464,   2121,    1848,     920,     224,       16;\n  5: 16, 215, 1270,  4330,   9400,   13503,   12890,    7960,     2960,      560,       32;\n  6: 19, 312, 2307, 10140,  29445,   59352,   84841,   86196,    61260,    29280,     8688,     1344,       64;\n  7: 22, 427, 3794, 20433,  74424,  193641,  370426,  527887,   561134,   440412,   249144,    97104,    23968,    3136,     128;\n  8: 25, 560, 5812, 37072, 162526,  518784, 1245748, 2292176,  3259921,  3586288,  3030832,  1936704,   909664,  299264,   63232,   7168,    256;\n  9: 28, 711, 8442, 62244, 319176, 1207962, 3494148, 7891380, 14090508, 20011383, 22616370, 20241072, 14187936, 7642656, 3068352, 873216, 161280, 16128, 512;\n","recurrence_sentence":"The row polynomials satisfy (3*n-2+(4*n-1)*x+2*x^2)*P(n,x) = (3*n+1+(13*n+6)*x+(18*n+13)*x^2+(8*n+12)*x^3+4*x^4)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((3*n + 1 + (13*n + 6)*x + (18*n + 13)*x^2 + (8*n + 12)*x^3 + 4*x^4)*P[n - 1, x])/(3*n - 2 + (4*n - 1)*x + 2*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((3*n + 1 + (13*n + 6)*x + (18*n + 13)*x^2 + (8*n + 12)*x^3 + 4*x^4)*v[i - 1])/(3*n - 2 + (4*n - 1)*x + 2*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-eed4f744ce063213","steps":["D+1"],"edges":["e-97f1ae3db6b837a6"],"text":"A200536 \u2192 D+1"},"headline":"A200536 \u2192 D+1","chain":null}