{"id":"n-077b2f078944fa2f","revision":0,"first_index":0,"fingerprint":"f6-dd008bbb0b25b199","name":"derivative of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*1^j.","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":true,"rows":[[],["2"],["52","8"],["1014","312","24"],["17576","8112","1248","64"],["285610","175760","40560","4160","160"],["4455516","3427320","1054560","162240","12480","384"],["67575326","62377224","23991240","4921280","567840","34944","896"],["1003976272","1081205216","499017792","127953280","19685120","1817088","93184","2048"],["14683152978","18071572896","9730846944","2994106752","575789760","70866432","5451264","239616","4608"],["212089987460","293663059560","180715728960","64872312960","14970533760","2303159040","236221440","15575040","599040","10240"],["3032886820678","4665979724120","3230293655160","1325248679040","356797721280","65870348544","8444916480","742410240","42831360","1464320","22528"],["43011849456888","72789283696272","55991756689440","25842349241280","7951492074240","1712629062144","263481394176","28953999360","2227230720","114216960","3514368","49152"],["605750213184506","1118308085879088","946260688051536","485261891308480","167975270068320","41347758786048","7421392602624","978645178368","94100497920","6434222080","296964096","8306688","106496"],["8480502984583084","16961005969166168","15656313202307232","8831766421814336","3396833239159360","940661512382592","192956207668224","29685570410496","3425258124288","292757104640","18015821824","755908608","19382272","229376"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","2","60","1350","27000","506250","9112500","159468750","2733750000","46132031250","768867187500","12686308593750","207594140625000","3373404785156250","54493461914062500"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["2"]],"recurrence":{"denominator":{"coeffs":[["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","0"],["13","2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":188815,"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-dfc8ba4c244d10ea"]},{"id":188816,"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-dfc8ba4c244d10ea"]},{"id":188817,"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-dfc8ba4c244d10ea"]},{"id":188818,"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-dfc8ba4c244d10ea"]}],"region_exact":{},"incoming_edges":[{"id":"e-dfc8ba4c244d10ea","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"0\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-077b2f078944fa2f","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":"adcdf4527904ddeb81f9460962297f2d75147c83c0ee5da8f0fca8111ca385a5","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-e88ccd05662b8cd9","role":"sequence","name":"basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*1^j.","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-c731c1158348989d","kind":"fingerprint_related","node_a_id":"n-e88ccd05662b8cd9","node_b_id":"n-077b2f078944fa2f","fingerprint":"f6-dd008bbb0b25b199","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-dd008bbb0b25b199"},"other_id":"n-e88ccd05662b8cd9","other":{"id":"n-e88ccd05662b8cd9","name":"basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*1^j.","first_index":0,"fingerprint":"f6-dd008bbb0b25b199","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["13","2"],["169","52","4"]],"card_rows":[["1"],["13","2"],["169","52","4"],["2197","1014","156","8"]],"row_sums_preview":["1","15","225","3375","50625","759375","11390625","170859375"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-81e6fb4f9f226d77","steps":["basis A013609"],"edges":["e-ec56a67a9fb88114"],"text":"A038327 \u2192 basis A013609"},"headline":"A038327 \u2192 basis A013609","chain":null}},{"id":"l-edb6984aa43e0e4b","kind":"fingerprint_related","node_a_id":"n-077b2f078944fa2f","node_b_id":"n-f5a389a0bc2a9f39","fingerprint":"f6-dd008bbb0b25b199","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-dd008bbb0b25b199"},"other_id":"n-f5a389a0bc2a9f39","other":{"id":"n-f5a389a0bc2a9f39","name":"reversal of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*1^j.","first_index":0,"fingerprint":"f6-dd008bbb0b25b199","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","13"],["4","52","169"]],"card_rows":[["1"],["2","13"],["4","52","169"],["8","156","1014","2197"]],"row_sums_preview":["1","15","225","3375","50625","759375","11390625","170859375"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-81e6fb4f9f226d77","steps":["basis A013609","reverse"],"edges":["e-ec56a67a9fb88114","e-df6e23293a0a08ae"],"text":"A038327 \u2192 basis A013609 \u2192 reverse"},"headline":"A038327 \u2192 basis A013609 \u2192 reverse","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:           0;\n  1:           2;\n  2:          52,           8;\n  3:        1014,         312,         24;\n  4:       17576,        8112,       1248,         64;\n  5:      285610,      175760,      40560,       4160,       160;\n  6:     4455516,     3427320,    1054560,     162240,     12480,      384;\n  7:    67575326,    62377224,   23991240,    4921280,    567840,    34944,     896;\n  8:  1003976272,  1081205216,  499017792,  127953280,  19685120,  1817088,   93184,   2048;\n  9: 14683152978, 18071572896, 9730846944, 2994106752, 575789760, 70866432, 5451264, 239616, 4608;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = (13*n+2*n*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[Together[((13*n + 2*n*x)*P[n - 1, x])/(n - 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = ((13*n + 2*n*x)*v[i - 1])/(n - 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-81e6fb4f9f226d77","steps":["basis A013609","D"],"edges":["e-ec56a67a9fb88114","e-dfc8ba4c244d10ea"],"text":"A038327 \u2192 basis A013609 \u2192 D"},"headline":"A038327 \u2192 basis A013609 \u2192 D","chain":null}