{"id":"n-9688148ffd278232","revision":0,"first_index":0,"fingerprint":"f6-d37b0b195ff0bdee","name":"one_plus_derivative of basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*5^j.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":2,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["896","1440","400"],["19456","53760","43200","8000"],["393216","1556480","2150400","1152000","160000"],["7602176","39321600","77824000","71680000","28800000","3200000"],["142606336","912261120","2359296000","3112960000","2150400000","691200000","64000000"],["2617245696","19964887040","63858278400","110100480000","108953600000","60211200000","16128000000","1280000000"],["47244640256","418759311360","1597190963200","3405774848000","4404019200000","3486515200000","1605632000000","368640000000","25600000000"],["841813590016","8504035246080","37688338022400","95831457792000","153259868160000","158544691200000","104595456000000","41287680000000","8294400000000","512000000000"],["14843406974976","168362718003200","850403524608000","2512555868160000","4791572889600000","6130394726400000","5284823040000000","2988441600000000","1032192000000000","184320000000000","10240000000000"],["259484744155136","3265549534494720","18519898980352000","62362925137920000","138190572748800000","210829207142400000","224781139968000000","166094438400000000","82182144000000000","25231360000000000","4055040000000000","204800000000000"],["4503599627370496","62276338597232640","391865944139366400","1481591918428160000","3741775508275200000","6633147491942400000","8433168285696000000","7706781941760000000","4982833152000000000","2191523840000000000","605552640000000000","88473600000000000","4096000000000000"],["77687093572141056","1170935903116328960","8095924017640243200","33961715158745088000","96303474697830400000","194572326430310400000","287436391317504000000","313231964897280000000","250470413107200000000","143948513280000000000","56979619840000000000","14313062400000000000","1916928000000000000","81920000000000000"],["1333065489701666816","21752386200199495680","163931026436286054400","755619574979756032000","2377320061112156160000","5392994583078502400000","9080041900081152000000","11497455652700160000000","10963118771404800000000","7792412852224000000000","4030558371840000000000","1450390323200000000000","333971456000000000000","41287680000000000000","1638400000000000000"],["22770199715985227776","399919646910500044800","3262857930029924352000","16393102643628605440000","56671468123481702400000","142639203666729369600000","269649729153925120000000","389144652860620800000000","431154586976256000000000","365437292380160000000000","233772385566720000000000","109924319232000000000000","36259758080000000000000","7707033600000000000000","884736000000000000000","32768000000000000000"],["387381625547900583936","7286463909115272888320","63987143505680007168000","348038179203191930880000","1311448211490288435200000","3626973959902828953600000","7607424195558899712000000","12326844761322291200000000","15565786114424832000000000","15329940870266880000000000","11693993356165120000000000","6800651216486400000000000","2931315179520000000000000","892547891200000000000000","176160768000000000000000","18874368000000000000000","655360000000000000000"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["2736","124416","5412096","228427776","9432723456","383113691136","15359376162816","609359740010496","23968149773746176","935976560656121856","36327590260465729536","1402560876143198601216","53903826104746713808896","2063356584060178007064576","78702315420581075412320256"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["896","1440","400"]],"recurrence":{"denominator":{"coeffs":[["9","5"],["5","0"]]},"terms":[{"coeff":{"coeffs":[["224","360","100"],["80","100","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":194707,"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-bd1a20fd2bff9cc0"]},{"id":194708,"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-bd1a20fd2bff9cc0"]},{"id":194709,"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-bd1a20fd2bff9cc0"]},{"id":194710,"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-bd1a20fd2bff9cc0"]}],"region_exact":{},"incoming_edges":[{"id":"e-bd1a20fd2bff9cc0","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-9688148ffd278232","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":"189a6339cb2f513656e852a44b90b38d2f1c47d6ca797aac41a6e340cf428892","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-93016690ff7736f4","role":"sequence","name":"basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*5^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-497896d103c3a879","kind":"fingerprint_related","node_a_id":"n-9bf2a4f2e5de2104","node_b_id":"n-9688148ffd278232","fingerprint":"f6-d37b0b195ff0bdee","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-d37b0b195ff0bdee"},"other_id":"n-9bf2a4f2e5de2104","other":{"id":"n-9bf2a4f2e5de2104","name":"one_plus_derivative of basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*5^j.","first_index":0,"fingerprint":"f6-d37b0b195ff0bdee","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["27","15"],["504","810","225"]],"card_rows":[["1"],["27","15"],["504","810","225"],["8208","22680","18225","3375"]],"row_sums_preview":["1","42","1539","52488","1712421","54206982","1678822119","51139504548"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a6a633b989da1879","steps":["basis A013610","D+1"],"edges":["e-cc1debec4a411a9c","e-e0a2a6c9a54f41d6"],"text":"A038271 \u2192 basis A013610 \u2192 D+1"},"headline":"A038271 \u2192 basis A013610 \u2192 D+1","chain":null}},{"id":"l-99f62e8ff0230a5a","kind":"fingerprint_related","node_a_id":"n-09e3e314b59270f4","node_b_id":"n-9688148ffd278232","fingerprint":"f6-d37b0b195ff0bdee","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-d37b0b195ff0bdee"},"other_id":"n-09e3e314b59270f4","other":{"id":"n-09e3e314b59270f4","name":"one_plus_derivative of basis_map_A013609 of Triangle of coefficients in expansion of (3+5x)^n.","first_index":0,"fingerprint":"f6-d37b0b195ff0bdee","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["18","10"],["224","360","100"]],"card_rows":[["1"],["18","10"],["224","360","100"],["2432","6720","5400","1000"]],"row_sums_preview":["1","28","684","15552","338256","7138368","147386304","2993075712"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c44f7114310405ad","steps":["basis A013609","D+1"],"edges":["e-c5caeaebd0a50686","e-5925b7a3cd3932dd"],"text":"A013622 \u2192 basis A013609 \u2192 D+1"},"headline":"A013622 \u2192 basis A013609 \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:             896,             1440,               400;\n  1:           19456,            53760,             43200,              8000;\n  2:          393216,          1556480,           2150400,           1152000,             160000;\n  3:         7602176,         39321600,          77824000,          71680000,           28800000,            3200000;\n  4:       142606336,        912261120,        2359296000,        3112960000,         2150400000,          691200000,           64000000;\n  5:      2617245696,      19964887040,       63858278400,      110100480000,       108953600000,        60211200000,        16128000000,         1280000000;\n  6:     47244640256,     418759311360,     1597190963200,     3405774848000,      4404019200000,      3486515200000,      1605632000000,       368640000000,       25600000000;\n  7:    841813590016,    8504035246080,    37688338022400,    95831457792000,    153259868160000,    158544691200000,    104595456000000,     41287680000000,     8294400000000,      512000000000;\n  8:  14843406974976,  168362718003200,   850403524608000,  2512555868160000,   4791572889600000,   6130394726400000,   5284823040000000,   2988441600000000,  1032192000000000,   184320000000000,   10240000000000;\n  9: 259484744155136, 3265549534494720, 18519898980352000, 62362925137920000, 138190572748800000, 210829207142400000, 224781139968000000, 166094438400000000, 82182144000000000, 25231360000000000, 4055040000000000, 204800000000000;\n","recurrence_sentence":"The row polynomials satisfy (5*n+9+5*x)*P(n,x) = (80*n+224+(100*n+360)*x+100*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 896 + 1440*x + 400*x^2.","mathematica":"P[0, x_] := 896 + 1440*x + 400*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((80*n + 224 + (100*n + 360)*x + 100*x^2)*P[n - 1, x])/(5*n + 9 + 5*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 896 + 1440*x + 400*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((80*n + 224 + (100*n + 360)*x + 100*x^2)*v[i - 1])/(5*n + 9 + 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-2bc1923905556bf6","steps":["basis A013611","D+1"],"edges":["e-fd91849598d32af1","e-bd1a20fd2bff9cc0"],"text":"A038319 \u2192 basis A013611 \u2192 D+1"},"headline":"A038319 \u2192 basis A013611 \u2192 D+1","chain":null}
