{"id":"n-22a40dfc2f208755","revision":0,"first_index":0,"fingerprint":"f6-0be541b3e25766a3","name":"one_plus_derivative of Triangle read by rows: T(n, k) is the Sheffer triangle ((1 - 3*x)^(-1/3), (-1/3)*log(1 - 3*x)). A generalized Stirling1 triangle.","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":false,"rows":[["1"],["2","1"],["9","7","1"],["67","63","15","1"],["698","736","225","26","1"],["9354","10684","3820","585","40","1"],["153304","186012","74289","13625","1260","57","1"],["2971016","3782596","1642977","342769","38570","2394","77","1"],["66468912","88052584","40887160","9411864","1220009","93212","4158","100","1"],["1686067120","2309908104","1133195580","282158148","40771458","3622437","200676","6750","126","1"],["47818487360","67440953152","34655608836","9205947420","1451217585","144862263","9401343","395490","10395","155","1"],["1499414132160","2169265498432","1160096571964","325481266836","55134857415","6043958283","443445849","21984303","726825","15345","187","1"],["51508352237440","76225583467008","42211610500344","12413471316664","2234683817586","264732930792","21444528237","1208059974","47300913","1262140","21879","222","1"],["1923770272081280","2905402053458176","1659394407242520","508467507629432","96460775793382","12203938375200","1073279944308","67049184345","2995896618","95066543","2091232","30303","260","1"],["77615376737203200","119399056828537600","70104277160823424","22276859908945824","4424324117140564","592425350404888","55867952609469","3803884958445","189186492066","6875719851","180506326","3330691","40950","301","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","17","146","1686","24484","428548","8780400","206138000","5455930400","160726884000","5215902457600","184881135443200","7106842376793600","294472188186713600"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["2","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-25","10","-1"],["30","-6","0"],["-9","0","0"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-7","2"],["6","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":185818,"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-17676725415a6a3d"]},{"id":185819,"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-17676725415a6a3d"]},{"id":185820,"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-17676725415a6a3d"]},{"id":185821,"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-17676725415a6a3d"]}],"region_exact":{},"incoming_edges":[{"id":"e-17676725415a6a3d","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-22a40dfc2f208755","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":"2abe31b6eda85da1ce592d6ca8ce73bdea50331d1d4f8536acee6d539fd28846","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-260e022530e5976c","role":"sequence","name":"Triangle read by rows: T(n, k) is the Sheffer triangle ((1 - 3*x)^(-1/3), (-1/3)*log(1 - 3*x)). A generalized Stirling1 triangle.","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:          2,          1;\n  2:          9,          7,          1;\n  3:         67,         63,         15,         1;\n  4:        698,        736,        225,        26,        1;\n  5:       9354,      10684,       3820,       585,       40,       1;\n  6:     153304,     186012,      74289,     13625,     1260,      57,      1;\n  7:    2971016,    3782596,    1642977,    342769,    38570,    2394,     77,    1;\n  8:   66468912,   88052584,   40887160,   9411864,  1220009,   93212,   4158,  100,   1;\n  9: 1686067120, 2309908104, 1133195580, 282158148, 40771458, 3622437, 200676, 6750, 126, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(9*n^2-30*n+25+(6*n-10)*x+x^2)*P(n-2,x) + (6*n-7+2*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + x;\nP[n_, x_] := P[n, x] = Expand[-(9*n^2 - 30*n + 25 + (6*n - 10)*x + x^2)*P[n - 2, x] + (6*n - 7 + 2*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = -(9*n^2 - 30*n + 25 + (6*n - 10)*x + x^2)*v[i - 2] + (6*n - 7 + 2*x)*v[i - 1]); 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-260e022530e5976c","steps":["D+1"],"edges":["e-17676725415a6a3d"],"text":"A286718 \u2192 D+1"},"headline":"A286718 \u2192 D+1","chain":null}