{"id":"n-f709fe4a87efd712","revision":0,"first_index":0,"fingerprint":"f6-99f5d32b38d6538a","name":"basis_map_A013610 of reversal of The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n>=2.","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":[["9","36","36"],["18","99","180","108"],["36","252","657","756","324"],["72","612","2070","3483","2916","972"],["144","1440","5976","13176","16281","10692","2916"],["288","3312","16272","44280","72090","70227","37908","8748"],["576","7488","42480","137376","277020","356724","286497","131220","26244"],["1152","16704","107424","402192","966168","1544508","1643166","1121931","446148","78732"],["2304","36864","264960","1126656","3138912","5987520","7919856","7173360","4258089","1495908","236196"],["4608","80640","640512","3048192","9657792","21391776","33802272","38106288","30036258","15766083","4960116","708588"],["9216","175104","1522944","8017920","28460160","71756928","131779872","177619392","174391380","121640940","57218481","16297524","2125764"],["18432","377856","3571200","20604672","80974080","228894336","478830528","750578400","881640936","766456020","479359782","204250491","53144100","6377292"],["36864","811008","8275968","51922944","223762176","700710912","1644344064","2937648384","4015017072","4177834848","3258087624","1846580328","719039673","172186884","19131876"],["73728","1732608","18984960","128673792","603293184","2072708352","5390820864","10808328960","16842979296","20400720912","19049679792","13467423528","6977820330","2501492787","554824404","57395628"],["147456","3686400","43167744","314302464","1592607744","5955296256","16999766784","37789120512","66110945472","91330379712","99301522320","84083886432","54357911244","25936446564","8614127169","1779264468","172186884"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["81","405","2025","10125","50625","253125","1265625","6328125","31640625","158203125","791015625","3955078125","19775390625","98876953125","494384765625"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013610","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["9","36","36"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2","3"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":201696,"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-a09d8f8afdbff0de"]},{"id":201697,"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-a09d8f8afdbff0de"]},{"id":201698,"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-a09d8f8afdbff0de"]}],"region_exact":{},"incoming_edges":[{"id":"e-a09d8f8afdbff0de","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-f709fe4a87efd712","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":"3eaeca39f3c95dc945e8508bbc0c8bcf884eaaaecb892c8501ef813d2111cac5","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-3a033458c346939f","role":"sequence","name":"reversal of The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n>=2.","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-7acb0973f72b8f53","kind":"fingerprint_related","node_a_id":"n-6433a9002006e547","node_b_id":"n-f709fe4a87efd712","fingerprint":"f6-99f5d32b38d6538a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-99f5d32b38d6538a"},"other_id":"n-6433a9002006e547","other":{"id":"n-6433a9002006e547","name":"mobius_x_over_x_plus_one of shift_x_plus_one of Riordan array (1-x,x/(1-x)).","first_index":0,"fingerprint":"f6-99f5d32b38d6538a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1","4","4"],["2","11","20","12"],["4","28","73","84","36"]],"card_rows":[["1","4","4"],["2","11","20","12"],["4","28","73","84","36"],["8","68","230","387","324","108"]],"row_sums_preview":["9","45","225","1125","5625","28125","140625","703125"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7717563017cb9e5f","steps":["x \u21a6 x+1","x \u21a6 x/(x+1)"],"edges":["e-34caf7309ed0b453","e-d2a1c5b6552f9304"],"text":"A159854 \u2192 x \u21a6 x+1 \u2192 x \u21a6 x/(x+1)"},"headline":"A159854 \u2192 x \u21a6 x+1 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-9d17ea6c3b19f9f4","kind":"fingerprint_related","node_a_id":"n-b438a1032ada5daf","node_b_id":"n-f709fe4a87efd712","fingerprint":"f6-99f5d32b38d6538a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-99f5d32b38d6538a"},"other_id":"n-b438a1032ada5daf","other":{"id":"n-b438a1032ada5daf","name":"basis_map_A013609 of basis_map_A007318 of The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n>=2.","first_index":0,"fingerprint":"f6-99f5d32b38d6538a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["16","16","4"],["48","80","44","8"],["144","336","292","112","16"]],"card_rows":[["16","16","4"],["48","80","44","8"],["144","336","292","112","16"],["432","1296","1548","920","272","32"]],"row_sums_preview":["36","180","900","4500","22500","112500","562500","2812500"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-f48bed27dbc29c5f","steps":["x \u21a6 x+1","basis A013609"],"edges":["e-3fc4d53dea6f488e","e-d3ff78d9b396025d"],"text":"A391533 \u2192 x \u21a6 x+1 \u2192 basis A013609"},"headline":"A391533 \u2192 x \u21a6 x+1 \u2192 basis A013609","chain":null}},{"id":"l-d6742356d79db929","kind":"fingerprint_related","node_a_id":"n-f709fe4a87efd712","node_b_id":"n-12545cb3cb924bbb","fingerprint":"f6-99f5d32b38d6538a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-99f5d32b38d6538a"},"other_id":"n-12545cb3cb924bbb","other":{"id":"n-12545cb3cb924bbb","name":"basis_map_A013610 of basis_map_A013609 of Riordan array (1-x,x/(1-x)).","first_index":0,"fingerprint":"f6-99f5d32b38d6538a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["9","36","36"],["36","198","360","216"],["144","1008","2628","3024","1296"]],"card_rows":[["9","36","36"],["36","198","360","216"],["144","1008","2628","3024","1296"],["576","4896","16560","27864","23328","7776"]],"row_sums_preview":["81","810","8100","81000","810000","8100000","81000000","810000000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7717563017cb9e5f","steps":["basis A013609","basis A013610"],"edges":["e-4ff089327fda7415","e-54abe09fe856b53f"],"text":"A159854 \u2192 basis A013609 \u2192 basis A013610"},"headline":"A159854 \u2192 basis A013609 \u2192 basis A013610","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:    9,    36,     36;\n  1:   18,    99,    180,     108;\n  2:   36,   252,    657,     756,     324;\n  3:   72,   612,   2070,    3483,    2916,      972;\n  4:  144,  1440,   5976,   13176,   16281,    10692,     2916;\n  5:  288,  3312,  16272,   44280,   72090,    70227,    37908,     8748;\n  6:  576,  7488,  42480,  137376,  277020,   356724,   286497,   131220,    26244;\n  7: 1152, 16704, 107424,  402192,  966168,  1544508,  1643166,  1121931,   446148,    78732;\n  8: 2304, 36864, 264960, 1126656, 3138912,  5987520,  7919856,  7173360,  4258089,  1495908,  236196;\n  9: 4608, 80640, 640512, 3048192, 9657792, 21391776, 33802272, 38106288, 30036258, 15766083, 4960116, 708588;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2+3*x)*P(n-1,x) for n >= 1, with P(0,x) = 9 + 36*x + 36*x^2.","mathematica":"P[0, x_] := 9 + 36*x + 36*x^2;\nP[n_, x_] := P[n, x] = Expand[(2 + 3*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] = 9 + 36*x + 36*x^2; for(n = 1, N, my(i = n + 1); v[i] = (2 + 3*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-f48bed27dbc29c5f","steps":["reverse","basis A013610"],"edges":["e-6248136e99299279","e-a09d8f8afdbff0de"],"text":"A391533 \u2192 reverse \u2192 basis A013610"},"headline":"A391533 \u2192 reverse \u2192 basis A013610","chain":null}
