{"id":"n-df88941ef8650983","revision":1,"first_index":0,"fingerprint":"f6-9ce300096a191a2b","name":"one_plus_derivative of Stirling2 triangle with scaled diagonals (powers of 2).","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"],["3","1"],["10","8","1"],["36","52","15","1"],["136","320","160","24","1"],["528","1936","1500","380","35","1"],["2080","11648","13216","5040","770","48","1"],["8256","69952","112560","60816","13720","1400","63","1"],["32896","419840","939520","693504","217056","32256","2352","80","1"],["131328","2519296","7742400","7625920","3186960","647472","68040","3720","99","1"],["524800","15116288","63264256","81818880","44508640","11886336","1689072","132000","5610","120","1"],["2098176","90698752","513934080","862797056","600131840","205173760","37887696","3970032","239580","8140","143","1"],["8390656","544194560","4157870080","8985360384","7887815936","3387850752","791300224","106803840","8583432","411840","11440","168","1"],["33558528","3265171456","33539251200","92714818560","101716975360","54124598528","15682867200","2652981760","272720448","17329312","676676","15652","195","1"],["134225920","19591036928","269963370496","950046289920","1292800104960","843221643264","298736325888","62098150400","7943066560","641921280","33041008","1070160","20930","224","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","4","19","104","641","4380","32803","266768","2337505","21925236","218946003","2316939256","25878593313","304020964876","3745210267939"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["3","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-8","0","1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","-4","2"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["6"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":165870,"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-9fe96d700c9448ac"]},{"id":165871,"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-9fe96d700c9448ac"]},{"id":165872,"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-9fe96d700c9448ac"]},{"id":165873,"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-9fe96d700c9448ac"]}],"region_exact":{},"incoming_edges":[{"id":"e-9fe96d700c9448ac","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-df88941ef8650983","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":"0f28d8490c59255b847d434c4ea55bbc3f6d73de03f117f4ef63211cade9c934","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-a7e9ebb9860a9dbe","role":"sequence","name":"Stirling2 triangle with scaled diagonals (powers of 2).","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-e52945bec715f8c5","kind":"oeis_match","node_a_id":"n-df88941ef8650983","node_b_id":null,"fingerprint":null,"oeis_id":"A383203","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A383203","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A383203","relation":"match_row_sums"}],"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:      3,       1;\n  2:     10,       8,       1;\n  3:     36,      52,      15,       1;\n  4:    136,     320,     160,      24,       1;\n  5:    528,    1936,    1500,     380,      35,      1;\n  6:   2080,   11648,   13216,    5040,     770,     48,     1;\n  7:   8256,   69952,  112560,   60816,   13720,   1400,    63,    1;\n  8:  32896,  419840,  939520,  693504,  217056,  32256,  2352,   80,  1;\n  9: 131328, 2519296, 7742400, 7625920, 3186960, 647472, 68040, 3720, 99, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(8-x^2)*P(n-2,x) - (4*x-2*x^2)*d/dx P(n-2,x) + 6*P(n-1,x) + 2*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + x;\nP[n_, x_] := P[n, x] = Expand[-(8 - x^2)*P[n - 2, x] - (4*x - 2*x^2)*D[P[n - 2, x], x] + 6*P[n - 1, x] + 2*x*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + x; for(n = 2, N, my(i = n + 1); v[i] = -(8 - x^2)*v[i - 2] - (4*x - 2*x^2)*deriv(v[i - 2], x) + 6*v[i - 1] + 2*x*deriv(v[i - 1], 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-a7e9ebb9860a9dbe","steps":["D+1"],"edges":["e-9fe96d700c9448ac"],"text":"A075497 \u2192 D+1"},"headline":"A075497 \u2192 D+1","chain":null}
