{"id":"n-563143c7a3f59683","revision":1,"first_index":0,"fingerprint":"f6-bb0a136ce054ce63","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.","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"],["2","1"],["6","6","1"],["22","33","13","1"],["92","184","116","24","1"],["422","1055","920","325","40","1"],["2074","6222","6945","3520","785","62","1"],["10754","37639","51261","34055","11291","1701","91","1"],["58202","232808","374500","308672","137326","31808","3388","128","1"],["326240","1468080","2724780","2685690","1507170","479220","80934","6306","174","1"],["1882960","9414800","19807800","22742400","15451800","6299160","1492470","189540","11100","230","1"],["11140560","61273080","144123320","189006840","151021200","75453840","23287440","4234560","414150","18645","297","1"],["67329992","403979952","1050619064","1549945760","1425339696","845463696","323657400","77841984","11108592","853072","30096","376","1"],["414499438","2694246347","7677014644","12586870725","13101963446","9012369795","4134563004","1247920245","238989036","27245933","1669954","46943","468","1"],["2593341586","18153391102","56245047313","101476199552","117997042163","92442536186","49550265765","18088347420","4396737345","681931250","63027965","3127124","71071","574","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","13","69","417","2763","19609","146793","1146833","9278595","77292261","659973933","5756169681","51137399979","461691066417"],"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"],["17","10"],["11","10"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["24","24","0"],["-12","-27","-15"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["12","30","24","6"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":164846,"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-0c759266bd791ca5"]},{"id":164847,"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-0c759266bd791ca5"]},{"id":164848,"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-0c759266bd791ca5"]}],"region_exact":{},"incoming_edges":[{"id":"e-0c759266bd791ca5","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-563143c7a3f59683","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":"0adaae8ee428ec05f95cab57f742bffea3d265b48167f391648d62ba96d4dd6a","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-047228a38014e6d4","role":"sequence","name":"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":[{"id":"l-34c5f745de974b1d","kind":"fingerprint_related","node_a_id":"n-563143c7a3f59683","node_b_id":"n-33bd4b527bca505e","fingerprint":"f6-bb0a136ce054ce63","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-bb0a136ce054ce63"},"other_id":"n-33bd4b527bca505e","other":{"id":"n-33bd4b527bca505e","name":"mobius_x_over_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.","first_index":0,"fingerprint":"f6-bb0a136ce054ce63","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","2"],["1","6","6"]],"card_rows":[["1"],["1","2"],["1","6","6"],["1","13","33","22"]],"row_sums_preview":["1","3","13","69","417","2763","19609","146793"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c963abfc0266d66c","steps":["\u2299 C(n+1,k+1) (shift +1)","x \u21a6 x/(x+1)"],"edges":["e-548c39a476f786f5","e-a929acd24243c7cc"],"text":"A108838 \u2192 \u2299 C(n+1,k+1) (shift +1) \u2192 x \u21a6 x/(x+1)"},"headline":"A108838 \u2192 \u2299 C(n+1,k+1) (shift +1) \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-6527bb7aee7df85d","kind":"oeis_match","node_a_id":"n-563143c7a3f59683","node_b_id":null,"fingerprint":null,"oeis_id":"A368708","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A368708","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A368708","relation":"match_row_sums"}],"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:      2,       1;\n  2:      6,       6,       1;\n  3:     22,      33,      13,       1;\n  4:     92,     184,     116,      24,       1;\n  5:    422,    1055,     920,     325,      40,      1;\n  6:   2074,    6222,    6945,    3520,     785,     62,     1;\n  7:  10754,   37639,   51261,   34055,   11291,   1701,    91,    1;\n  8:  58202,  232808,  374500,  308672,  137326,  31808,  3388,  128,   1;\n  9: 326240, 1468080, 2724780, 2685690, 1507170, 479220, 80934, 6306, 174, 1;\n","recurrence_sentence":"The row polynomials satisfy (n^2+7*n+12)*P(n,x) = (11*n^2+17*n+12+(10*n^2+10*n)*x)*P(n-1,x) - (12*n-24+(27*n-24)*x+15*n*x^2)*d/dx P(n-1,x) + (12+30*x+24*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[((11*n^2 + 17*n + 12 + (10*n^2 + 10*n)*x)*P[n - 1, x] - (12*n - 24 + (27*n - 24)*x + 15*n*x^2)*D[P[n - 1, x], x] + (12 + 30*x + 24*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] = ((11*n^2 + 17*n + 12 + (10*n^2 + 10*n)*x)*v[i - 1] - (12*n - 24 + (27*n - 24)*x + 15*n*x^2)*deriv(v[i - 1], x) + (12 + 30*x + 24*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-c963abfc0266d66c","steps":["\u2299 C(n+1,k+1) (shift +1)","x \u21a6 x+1"],"edges":["e-548c39a476f786f5","e-0c759266bd791ca5"],"text":"A108838 \u2192 \u2299 C(n+1,k+1) (shift +1) \u2192 x \u21a6 x+1"},"headline":"A108838 \u2192 \u2299 C(n+1,k+1) (shift +1) \u2192 x \u21a6 x+1","chain":null}