{"id":"n-41b6e8702634279f","revision":0,"first_index":0,"fingerprint":"f6-2fc5396437dfa965","name":"shift_x_plus_one of basis_map_A013611 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":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["1"],["17","4"],["289","136","16"],["4913","3468","816","64"],["83521","78608","27744","4352","256"],["1419857","1670420","786080","184960","21760","1024"],["24137569","34076568","20045040","6288640","1109760","104448","4096"],["410338673","675851932","477071952","187087040","44020480","6214656","487424","16384"],["6975757441","13130837536","10813630912","5088767488","1496696320","281731072","33144832","2228224","65536"],["118587876497","251127267876","236355075648","129763570944","45798907392","10776213504","1690386432","170459136","10027008","262144"],["2015993900449","4743515059880","5022545357520","3151401008640","1297635709440","366391259136","71841423360","9659351040","852295680","44564480","1048576"],["34271896307633","88703731619756","104357331317360","73663998576960","34665411095040","11419194243072","2686869233664","451574661120","53126430720","4166778880","196083712","4194304"],["582622237229761","1645051022766384","2128889558874144","1669717301077760","883967982923520","332787946512384","91353553944576","18424246173696","2709447966720","283340963840","20000538624","855638016","16777216"],["9904578032905937","30296356335947572","42771326591925984","36900752353818496","21706324914010880","9193267022404608","2884162203107328","678626400731136","119757600129024","15654588252160","1473373011968","94548000768","3707764736","67108864"],["168377826559400929","554656369842732472","848297977406532016","798398096382618368","516610532953458944","243110839036921856","85803825542443008","23073297624858624","4750384805117952","745158400802816","87665694212096","7500808060928","441224003584","15971909632","268435456"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","21","441","9261","194481","4084101","85766121","1801088541","37822859361","794280046581","16679880978201","350277500542221","7355827511386641","154472377739119461","3243919932521508681"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["17","4"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":189500,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, -1]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, -1]","region_exact":null,"edges":["e-519e8781acca086b"]},{"id":189501,"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-519e8781acca086b","e-a64dc177ad002567","e-a85a1f99c5538e68"]},{"id":189502,"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-519e8781acca086b","e-a64dc177ad002567","e-a85a1f99c5538e68"]},{"id":189503,"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-519e8781acca086b"]},{"id":189504,"position":4,"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-a64dc177ad002567","e-a85a1f99c5538e68"]}],"region_exact":{},"incoming_edges":[{"id":"e-a64dc177ad002567","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-41b6e8702634279f","premise_count":1,"theorem":"geometric family P_k = g(t) (a + d t)^k: T(p) = g(t) p(a + d t), so the roots of T(p) are those of g and the points (r - a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"09707fba82ec2bcd16c71acf686b9776747e2160c47240b3cece0c497681e123","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-0088aa50f9dba3f3","role":"sequence","name":"basis_map_A013611 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]"}]},{"id":"e-a85a1f99c5538e68","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-41b6e8702634279f","premise_count":1,"theorem":"geometric family P_k = g(t) (a + d t)^k: T(p) = g(t) p(a + d t), so the roots of T(p) are those of g and the points (r - a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"530dc31b3c261e826693d7ecb932fe9757967509aa19b9cf35cff16b5d0dc7c1","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-e91c3b4104e64a83","role":"sequence","name":"basis_map_A013609 of Triangle of coefficients in expansion of (14 + x)^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]"}]},{"id":"e-519e8781acca086b","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-41b6e8702634279f","premise_count":1,"theorem":"a real affine substitution x -> a x + b (a != 0), the reversal x^d p(1/x) (d >= deg p) and a real Mobius map (c x + e)^d p((a x + b)/(c x + e)) (a e - b c != 0, d >= deg p) map real roots to real roots: a root r of p gives the root (r - b)/a, 1/r (and 0) or (e r - b)/(a - c r) (and -e/c); for an input row that is a polynomial the degree condition holds automatically","lean_route":null,"lean_declarations":["Polynomial.Splits.comp_of_degree_le_one"],"certificate_hash":"6f8e95ba68aa123a5d4a5571e36b339a8061dfab4f2b67b887e84e3926013c25","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-0088aa50f9dba3f3","role":"sequence","name":"basis_map_A013611 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":[],"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:           17,            4;\n  2:          289,          136,           16;\n  3:         4913,         3468,          816,           64;\n  4:        83521,        78608,        27744,         4352,         256;\n  5:      1419857,      1670420,       786080,       184960,       21760,        1024;\n  6:     24137569,     34076568,     20045040,      6288640,     1109760,      104448,       4096;\n  7:    410338673,    675851932,    477071952,    187087040,    44020480,     6214656,     487424,     16384;\n  8:   6975757441,  13130837536,  10813630912,   5088767488,  1496696320,   281731072,   33144832,   2228224,    65536;\n  9: 118587876497, 251127267876, 236355075648, 129763570944, 45798907392, 10776213504, 1690386432, 170459136, 10027008, 262144;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (17+4*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(17 + 4*x)*P[n - 1, 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] = (17 + 4*x)*v[i - 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 A013611","x \u21a6 x+1"],"edges":["e-950a320de9db6fb9","e-519e8781acca086b"],"text":"A038327 \u2192 basis A013611 \u2192 x \u21a6 x+1"},"headline":"A038327 \u2192 basis A013611 \u2192 x \u21a6 x+1","chain":null}