{"id":"n-8010313896479d44","revision":0,"first_index":0,"fingerprint":"f6-dd67c79f07eb5aef","name":"shift_x_plus_one of shift_x_plus_one of Array read by antidiagonals: number of antichains (or order ideals) in the poset 3*m*n or plane partitions with rows <= m, columns <= n and entries <= 3.","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"],["3","1"],["13","8","1"],["69","62","16","1"],["417","492","194","28","1"],["2763","4035","2145","495","45","1"],["19609","34128","22850","7300","1110","68","1"],["146793","296548","239568","98084","21196","2268","98","1"],["1146833","2637400","2496088","1248352","351736","54880","4312","136","1"],["9278595","23930637","25968414","15355788","5350296","1107120","130032","7734","183","1"],["77292261","220933080","270423915","184665540","76774740","19885152","3149580","286740","13215","240","1"],["659973933","2070744786","2822327640","2186382825","1057214070","329721216","66154572","8254950","595485","21670","308","1"],["5756169681","19667422404","29539800378","25600971924","14123521863","5159416680","1257744708","201157704","20203095","1174932","34298","388","1"],["51137399979","188995757535","310148183007","297367178823","184388100039","77285334555","22223168253","4356944163","567328905","46631013","2217501","52637","481","1"],["461691066417","1835169778848","3266861699556","3433581852792","2364554236044","1119124862856","371306829894","86464354548","13922274366","1500250752","102282180","4025112","78624","588","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","4","22","148","1132","9484","85066","804556","7939738","81128800","853424464","9201391456","101327618056","1136518296892","12954283592578"],"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":{"denominator":{"coeffs":[["12"],["7"],["1"]]},"terms":[{"coeff":{"coeffs":[["12","0"],["27","10"],["21","10"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["48","24","0"],["-54","-57","-15"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["72","96","42","6"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":179405,"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-c3c466da8e0e3b10"]},{"id":179406,"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-c3c466da8e0e3b10"]},{"id":179407,"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-c3c466da8e0e3b10"]}],"region_exact":{},"incoming_edges":[{"id":"e-c3c466da8e0e3b10","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-8010313896479d44","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":"33fb9249c2a26d3f369c131d7de6cb297b4d78d379bbbb886c602af56097cd38","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}}],"premises":[{"position":0,"node_id":"n-563143c7a3f59683","role":"sequence","name":"shift_x_plus_one of Array read by antidiagonals: number of antichains (or order ideals) in the poset 3*m*n or plane partitions with rows <= m, columns <= n and entries <= 3.","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:       3,        1;\n  2:      13,        8,        1;\n  3:      69,       62,       16,        1;\n  4:     417,      492,      194,       28,       1;\n  5:    2763,     4035,     2145,      495,      45,       1;\n  6:   19609,    34128,    22850,     7300,    1110,      68,      1;\n  7:  146793,   296548,   239568,    98084,   21196,    2268,     98,    1;\n  8: 1146833,  2637400,  2496088,  1248352,  351736,   54880,   4312,  136,   1;\n  9: 9278595, 23930637, 25968414, 15355788, 5350296, 1107120, 130032, 7734, 183, 1;\n","recurrence_sentence":"The row polynomials satisfy (n^2+7*n+12)*P(n,x) = (21*n^2+27*n+12+(10*n^2+10*n)*x)*P(n-1,x) - (54*n-48+(57*n-24)*x+15*n*x^2)*d/dx P(n-1,x) + (72+96*x+42*x^2+6*x^3)*d^2/dx^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[((21*n^2 + 27*n + 12 + (10*n^2 + 10*n)*x)*P[n - 1, x] - (54*n - 48 + (57*n - 24)*x + 15*n*x^2)*D[P[n - 1, x], x] + (72 + 96*x + 42*x^2 + 6*x^3)*D[P[n - 1, x], {x, 2}])/(n^2 + 7*n + 12)]];\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] = ((21*n^2 + 27*n + 12 + (10*n^2 + 10*n)*x)*v[i - 1] - (54*n - 48 + (57*n - 24)*x + 15*n*x^2)*deriv(v[i - 1], x) + (72 + 96*x + 42*x^2 + 6*x^3)*deriv(deriv(v[i - 1], x), x))/(n^2 + 7*n + 12)); 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-047228a38014e6d4","steps":["x \u21a6 x+1","x \u21a6 x+1"],"edges":["e-0c759266bd791ca5","e-c3c466da8e0e3b10"],"text":"A056939 \u2192 x \u21a6 x+1 \u2192 x \u21a6 x+1"},"headline":"A056939 \u2192 x \u21a6 x+1 \u2192 x \u21a6 x+1","chain":null}
