{"id":"n-1f64b8c87ab9b182","revision":1,"first_index":1,"fingerprint":"f6-d99d39db05a8c355","name":"one_plus_derivative of Triangle read by rows: 3-Stirling numbers of the second kind.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["4","1"],["16","9","1"],["64","61","15","1"],["256","369","151","22","1"],["1024","2101","1275","305","30","1"],["4096","11529","9751","3410","545","39","1"],["16384","61741","70035","33621","7770","896","49","1"],["65536","325089","481951","305382","95781","15834","1386","60","1"],["262144","1690981","3216795","2619625","1071630","238287","29694","2046","72","1"],["1048576","8717049","20991751","21554170","11192665","3216213","535227","52200","2910","85","1"],["4194304","44633821","134667555","171870941","111095490","40138582","8568483","1109427","87120","4015","99","1"],["16777216","227363409","852639151","1337764142","1060634861","472342728","125823412","20772180","2154867","139315","5401","114","1"],["67108864","1153594261","5343198315","10216988145","9822843030","5311719413","1730576848","354317392","46630584","3965962","214929","7111","130","1"],["268435456","5835080169","33212784151","76862115330","88799732385","57628317747","22617487893","5628068160","913884400","98188090","6974968","321594","9191","147","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","26","141","799","4736","29371","190497","1291020","9131275","67310847","516369838","4116416797","34051164985","291871399682"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["4","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":165844,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-bbdbb484d8de66ad"]},{"id":165845,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-bbdbb484d8de66ad"]},{"id":165846,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-bbdbb484d8de66ad"]},{"id":165847,"position":3,"claim":"interlacing","from_index":2,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2","region_exact":null,"edges":["e-bbdbb484d8de66ad"]}],"region_exact":{},"incoming_edges":[{"id":"e-bbdbb484d8de66ad","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-1f64b8c87ab9b182","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":"647f5ae1d13ed19dd91b1345f6338af26f196bb40f1f99e4762c33a1f7efbde7","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-a2b8ff03da0c14a3","role":"sequence","name":"Triangle read by rows: 3-Stirling numbers of the second kind.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-1af1db90c24f2d9c","kind":"fingerprint_related","node_a_id":"n-41442279a7af7481","node_b_id":"n-1f64b8c87ab9b182","fingerprint":"f6-d99d39db05a8c355","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-d99d39db05a8c355"},"other_id":"n-41442279a7af7481","other":{"id":"n-41442279a7af7481","name":"basis_map_A008277 of Cube of lower triangular normalized binomial matrix.","first_index":0,"fingerprint":"f6-d99d39db05a8c355","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["4","1"],["16","9","1"]],"card_rows":[["1"],["4","1"],["16","9","1"],["64","61","15","1"]],"row_sums_preview":["1","5","26","141","799","4736","29371","190497"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-8d887bfc40d036b9","steps":["basis A008277"],"edges":["e-6d94a10ae1f2ed2c"],"text":"A027465 \u2192 basis A008277"},"headline":"A027465 \u2192 basis A008277","chain":null}},{"id":"l-cd67789993b0e028","kind":"fingerprint_related","node_a_id":"n-1f64b8c87ab9b182","node_b_id":"n-329498b88a584186","fingerprint":"f6-d99d39db05a8c355","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-d99d39db05a8c355"},"other_id":"n-329498b88a584186","other":{"id":"n-329498b88a584186","name":"basis_map_A008277 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.","first_index":0,"fingerprint":"f6-d99d39db05a8c355","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["256","144","16"],["4096","3904","960","64"],["65536","94464","38656","5632","256"]],"card_rows":[["256","144","16"],["4096","3904","960","64"],["65536","94464","38656","5632","256"],["1048576","2151424","1305600","312320","30720","1024"]],"row_sums_preview":["416","9024","204544","4849664","120303616","3121102848","84608286720","2393708953600"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a93042cd70c4761e","steps":["basis A008277"],"edges":["e-1e482bc7f83a739a"],"text":"A038330 \u2192 basis A008277"},"headline":"A038330 \u2192 basis A008277","chain":null}},{"id":"l-421f94ec900422d2","kind":"oeis_match","node_a_id":"n-1f64b8c87ab9b182","node_b_id":null,"fingerprint":null,"oeis_id":"A143496","in_catalog":0,"target":"triangle","row_alignment":"tabl","rows_matched":9,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":4,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":9,"target":"triangle","terms_compared":48},"in_catalog":false,"oeis_id":"A143496","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-fb873dcf3a603e68","kind":"oeis_match","node_a_id":"n-1f64b8c87ab9b182","node_b_id":null,"fingerprint":null,"oeis_id":"A045379","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A045379","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A045379","relation":"match_row_sums"},{"oeis_id":"A143496","relation":"match_triangle"}],"bfiles":[{"node_id":"n-1f64b8c87ab9b182","oeis_id":"A045379","target":"row_sums","oeis_offset":0,"first_row":1,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":21,"oeis_terms_available":21,"term_count":200,"raw_size":26345},{"node_id":"n-1f64b8c87ab9b182","oeis_id":"A143496","target":"triangle","oeis_offset":4,"first_row":1,"rows_used":44,"terms_matched":48,"oeis_terms_agreeing":48,"oeis_terms_available":48,"term_count":990,"raw_size":24150}],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:      1;\n   2:      4,       1;\n   3:     16,       9,       1;\n   4:     64,      61,      15,       1;\n   5:    256,     369,     151,      22,       1;\n   6:   1024,    2101,    1275,     305,      30,      1;\n   7:   4096,   11529,    9751,    3410,     545,     39,     1;\n   8:  16384,   61741,   70035,   33621,    7770,    896,    49,    1;\n   9:  65536,  325089,  481951,  305382,   95781,  15834,  1386,   60,  1;\n  10: 262144, 1690981, 3216795, 2619625, 1071630, 238287, 29694, 2046, 72, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (4+x)*P(n-1,x) + x*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(4 + x)*P[n - 1, x] + x*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (4 + x)*v[i - 1] + 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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}},{"oeis_id":"A143496","target":"triangle","first_row":1,"oeis_offset":4,"reversed":false,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   4:      1;\n   5:      4,       1;\n   6:     16,       9,       1;\n   7:     64,      61,      15,       1;\n   8:    256,     369,     151,      22,       1;\n   9:   1024,    2101,    1275,     305,      30,      1;\n  10:   4096,   11529,    9751,    3410,     545,     39,     1;\n  11:  16384,   61741,   70035,   33621,    7770,    896,    49,    1;\n  12:  65536,  325089,  481951,  305382,   95781,  15834,  1386,   60,  1;\n  13: 262144, 1690981, 3216795, 2619625, 1071630, 238287, 29694, 2046, 72, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (4+x)*P(n-1,x) + x*d/dx P(n-1,x) for n >= 5, with P(4,x) = 1.","mathematica":"P[4, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(4 + x)*P[n - 1, x] + x*D[P[n - 1, x], x]];\nTable[PadRight[CoefficientList[P[n, x], x], n - 3], {n, 4, 13}] // Flatten","pari":"P(N) = {my(v = vector(N - 3)); v[1] = 1; for(n = 5, N, my(i = n - 3); v[i] = (4 + x)*v[i - 1] + x*deriv(v[i - 1], x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - 3 - #c)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i + 3, v[i])))};\nT(13)","reading":{"first_row":1,"offset":4,"reversed":false,"trim_zeros":"none","regular_triangle":true}}],"derivation":{"root":"n-a2b8ff03da0c14a3","steps":["D+1"],"edges":["e-bbdbb484d8de66ad"],"text":"A143495 \u2192 D+1"},"headline":"A143495 \u2192 D+1","chain":null}
