{"id":"n-b99042e296c8e644","revision":0,"first_index":0,"fingerprint":"f6-951a7e0f7b8ffcff","name":"one_plus_derivative of Triangle of coefficients in expansion of (3+5x)^n.","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":true,"rows":[["1"],["8","5"],["39","80","25"],["162","585","600","125"],["621","3240","5850","4000","625"],["2268","15525","40500","48750","25000","3125"],["8019","68040","232875","405000","365625","150000","15625"],["27702","280665","1190700","2716875","3543750","2559375","875000","78125"],["94041","1108080","5613300","15876000","27168750","28350000","17062500","5000000","390625"],["314928","4231845","24931800","84199500","178605000","244518750","212625000","109687500","28125000","1953125"],["1043199","15746400","105796125","415530000","1052493750","1786050000","2037656250","1518750000","685546875","156250000","9765625"],["3424842","57375945","433026000","1939595625","5713537500","11577431250","16372125000","16010156250","10441406250","4189453125","859375000","48828125"],["11160261","205490520","1721278350","8660520000","29093934375","68562450000","115774312500","140332500000","120076171875","69609375000","25136718750","4687500000","244140625"],["36137988","725416965","6678441900","37294364250","140733450000","378221146875","742759875000","1075047187500","1140201562500","867216796875","452460937500","148535156250","25390625000","1220703125"],["116385579","2529659160","25389593775","155830311000","652651374375","1970268300000","4412580046875","7427598750000","9406662890625","8868234375000","6070517578125","2879296875000","866455078125","136718750000","6103515625"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","13","144","1472","14336","135168","1245184","11272192","100663296","889192448","7784628224","67645734912","584115552256","5016521801728","42880953483264"],"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","5"],["5","0"]]},"terms":[{"coeff":{"coeffs":[["9","30","25"],["15","25","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":183649,"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-0bf245e425ee01ec"]},{"id":183650,"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-0bf245e425ee01ec"]},{"id":183651,"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-0bf245e425ee01ec"]},{"id":183652,"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-0bf245e425ee01ec"]}],"region_exact":{},"incoming_edges":[{"id":"e-0bf245e425ee01ec","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-b99042e296c8e644","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":"8f43a0dbb59c6ff095cbf057ce027c014484a090a6e50de75f6609352d64f67b","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-c44f7114310405ad","role":"sequence","name":"Triangle of coefficients in expansion of (3+5x)^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":[{"id":"l-35337b10cd419f17","kind":"fingerprint_related","node_a_id":"n-b99042e296c8e644","node_b_id":"n-2ae0099201780429","fingerprint":"f6-951a7e0f7b8ffcff","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-951a7e0f7b8ffcff"},"other_id":"n-2ae0099201780429","other":{"id":"n-2ae0099201780429","name":"one_plus_derivative of basis_map_A013609 of Triangle of coefficients in expansion of (1+5x)^n.","first_index":0,"fingerprint":"f6-951a7e0f7b8ffcff","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["156","320","100"],["1296","4680","4800","1000"],["9936","51840","93600","64000","10000"]],"card_rows":[["156","320","100"],["1296","4680","4800","1000"],["9936","51840","93600","64000","10000"],["72576","496800","1296000","1560000","800000","100000"]],"row_sums_preview":["576","11776","229376","4325376","79691776","1442840576","25769803776","455266533376"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-bbfc5a54ff13a785","steps":["basis A013609","D+1"],"edges":["e-31086658f6c15e80","e-e1d4711e5a448783"],"text":"A013612 \u2192 basis A013609 \u2192 D+1"},"headline":"A013612 \u2192 basis A013609 \u2192 D+1","chain":null}},{"id":"l-ff6d32e83a256572","kind":"fingerprint_related","node_a_id":"n-b99042e296c8e644","node_b_id":"n-3c7b6c13e5ec85c4","fingerprint":"f6-951a7e0f7b8ffcff","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-951a7e0f7b8ffcff"},"other_id":"n-3c7b6c13e5ec85c4","other":{"id":"n-3c7b6c13e5ec85c4","name":"one_plus_derivative of basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*5^j.","first_index":0,"fingerprint":"f6-951a7e0f7b8ffcff","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["32","20"],["624","1280","400"]],"card_rows":[["1"],["32","20"],["624","1280","400"],["10368","37440","38400","8000"]],"row_sums_preview":["1","52","2304","94208","3670016","138412032","5100273664","184683593728"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a6a633b989da1879","steps":["basis A013611","D+1"],"edges":["e-62a1b0a86f60f435","e-991a184d1289ea8c"],"text":"A038271 \u2192 basis A013611 \u2192 D+1"},"headline":"A038271 \u2192 basis A013611 \u2192 D+1","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:      1;\n  1:      8,       5;\n  2:     39,      80,       25;\n  3:    162,     585,      600,      125;\n  4:    621,    3240,     5850,     4000,       625;\n  5:   2268,   15525,    40500,    48750,     25000,      3125;\n  6:   8019,   68040,   232875,   405000,    365625,    150000,     15625;\n  7:  27702,  280665,  1190700,  2716875,   3543750,   2559375,    875000,     78125;\n  8:  94041, 1108080,  5613300, 15876000,  27168750,  28350000,  17062500,   5000000,   390625;\n  9: 314928, 4231845, 24931800, 84199500, 178605000, 244518750, 212625000, 109687500, 28125000, 1953125;\n","recurrence_sentence":"The row polynomials satisfy (5*n-2+5*x)*P(n,x) = (15*n+9+(25*n+30)*x+25*x^2)*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[((15*n + 9 + (25*n + 30)*x + 25*x^2)*P[n - 1, x])/(5*n - 2 + 5*x)]];\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] = ((15*n + 9 + (25*n + 30)*x + 25*x^2)*v[i - 1])/(5*n - 2 + 5*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-c44f7114310405ad","steps":["D+1"],"edges":["e-0bf245e425ee01ec"],"text":"A013622 \u2192 D+1"},"headline":"A013622 \u2192 D+1","chain":null}
