{"id":"n-2de3c606f08cd985","revision":0,"first_index":0,"fingerprint":"f6-ade2b5b96bd8d678","name":"basis_map_A013609 of basis_map_A013611 of One half of odd-numbered entries of even-numbered rows of Pascal's triangle A007318.","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":false,"rows":[["1"],["12","16"],["128","320","192"],["1344","4864","5632","2048"],["14080","66560","113664","81920","20480"],["147456","860160","1941504","2097152","1064960","196608"],["1544192","10715136","30081024","43384832","33488896","12845056","1835008"],["16171008","130088960","436862976","789053440","820510720","484442112","146800640","16777216"],["169345024","1549533184","6057885696","13153337344","17233346560","13816037376","6517948416","1610612736","150994944"],["1773404160","18187550720","81136713728","205856440320","325666734080","330762813440","213406187520","82946555904","17112760320","1342177280"],["18571329536","210994462720","1057706999808","3071136497664","5696452034560","7014017138688","5761631518720","3082712776704","1009854185472","177167400960","11811160064"],["194481487872","2424576147456","13492706869248","44126493999104","93917378772992","136001750040576","135937862402048","93024696664064","42201274908672","11858404704256","1795296329728","103079215104"],["2036636581888","27642141081600","169090178088960","615066160332800","1477643232870400","2461349120573440","2901626218086400","2424337239900160","1412614743654400","552762290995200","135119671132160","17867063951360","893353197568"],["21327935176704","313051844706304","2087853395804160","8361506756362240","22382470443827200","42167645314744320","57311270503710720","56625536025231360","40411948583485440","20390786734489600","6977088473006080","1500833371914240","175097226723328","7696581394432"],["223349036810240","3525250100756480","25458185645064192","111311808416972800","328545456042803200","690816283783987200","1063896040970649600","1214466942849515520","1027488462195916800","636871494546227200","282031604472217600","85333097431695360","16309055974801408","1693247906775040","65970697666560"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","28","640","13888","296704","6307840","133894144","2840707072","60259041280","1278191337472","27112055504896","575078001541120","12198048050446336","258734113190576128","5488036250071859200"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013609","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["12","16"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-16","-64","-64"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["12","16"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":200485,"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-558cf966b6b39f46"]},{"id":200486,"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-558cf966b6b39f46"]},{"id":200487,"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-558cf966b6b39f46"]},{"id":200488,"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-558cf966b6b39f46"]}],"region_exact":{},"incoming_edges":[{"id":"e-558cf966b6b39f46","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-2de3c606f08cd985","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":"112d38565ddcd3a317d610245f29e8edc9a1a8ed5d132431419e34d24f5ef70a","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-55371dcfaf771579","role":"sequence","name":"basis_map_A013611 of One half of odd-numbered entries of even-numbered rows of Pascal's triangle A007318.","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-0562b215899be40f","kind":"fingerprint_related","node_a_id":"n-1ebae075a566c573","node_b_id":"n-2de3c606f08cd985","fingerprint":"f6-ade2b5b96bd8d678","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-ade2b5b96bd8d678"},"other_id":"n-1ebae075a566c573","other":{"id":"n-1ebae075a566c573","name":"reversal of basis_map_A013609 of Even row Pascal-square read by antidiagonals.","first_index":0,"fingerprint":"f6-ade2b5b96bd8d678","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["4","3"],["12","20","8"]],"card_rows":[["1"],["4","3"],["12","20","8"],["32","88","76","21"]],"row_sums_preview":["1","7","40","217","1159","6160","32689","173383"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-79ebafb266da3b41","steps":["basis A013609","reverse"],"edges":["e-1a56a7670c2d3c41","e-6d67f02ade321477"],"text":"A172431 \u2192 basis A013609 \u2192 reverse"},"headline":"A172431 \u2192 basis A013609 \u2192 reverse","chain":null}},{"id":"l-6fa0b5a6b024ccef","kind":"fingerprint_related","node_a_id":"n-7c7a913d07444057","node_b_id":"n-2de3c606f08cd985","fingerprint":"f6-ade2b5b96bd8d678","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-ade2b5b96bd8d678"},"other_id":"n-7c7a913d07444057","other":{"id":"n-7c7a913d07444057","name":"basis_map_A013609 of Even row Pascal-square read by antidiagonals.","first_index":0,"fingerprint":"f6-ade2b5b96bd8d678","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["3","4"],["8","20","12"]],"card_rows":[["1"],["3","4"],["8","20","12"],["21","76","88","32"]],"row_sums_preview":["1","7","40","217","1159","6160","32689","173383"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-79ebafb266da3b41","steps":["basis A013609"],"edges":["e-1a56a7670c2d3c41"],"text":"A172431 \u2192 basis A013609"},"headline":"A172431 \u2192 basis A013609","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:         12,          16;\n  2:        128,         320,         192;\n  3:       1344,        4864,        5632,         2048;\n  4:      14080,       66560,      113664,        81920,        20480;\n  5:     147456,      860160,     1941504,      2097152,      1064960,       196608;\n  6:    1544192,    10715136,    30081024,     43384832,     33488896,     12845056,      1835008;\n  7:   16171008,   130088960,   436862976,    789053440,    820510720,    484442112,    146800640,    16777216;\n  8:  169345024,  1549533184,  6057885696,  13153337344,  17233346560,  13816037376,   6517948416,  1610612736,   150994944;\n  9: 1773404160, 18187550720, 81136713728, 205856440320, 325666734080, 330762813440, 213406187520, 82946555904, 17112760320, 1342177280;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(16+64*x+64*x^2)*P(n-2,x) + (12+16*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 12 + 16*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 12 + 16*x;\nP[n_, x_] := P[n, x] = Expand[-(16 + 64*x + 64*x^2)*P[n - 2, x] + (12 + 16*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; v[2] = 12 + 16*x; for(n = 2, N, my(i = n + 1); v[i] = -(16 + 64*x + 64*x^2)*v[i - 2] + (12 + 16*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-2245b35ec7cd4857","steps":["basis A013611","basis A013609"],"edges":["e-61d2480a3633cd97","e-558cf966b6b39f46"],"text":"A091044 \u2192 basis A013611 \u2192 basis A013609"},"headline":"A091044 \u2192 basis A013611 \u2192 basis A013609","chain":null}
