{"id":"n-a882d0a2ea16bc46","revision":0,"first_index":1,"fingerprint":"f6-c277ed2a39f7c92c","name":"reversal of basis_map_A007318 of T(n,k) = binomial(n,k)*binomial(n+k,k), 0 <= k <= n, triangle read by rows.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["2","3"],["6","18","13"],["20","90","132","63"],["70","420","930","900","321"],["252","1890","5600","8190","5910","1683"],["924","8316","30870","60480","65940","37926","8989"],["3432","36036","160776","395010","577080","501228","239624","48639"],["12870","154440","804804","2378376","4358970","5072760","3660300","1497096","265729"],["48620","656370","3912480","13513500","29801772","43513470","42060480","25951860","9274410","1462563"],["184756","2771340","18597150","73513440","189549360","333080748","403933530","333796320","179874090","57074490","8097453"],["705432","11639628","86835320","386601930","1141208640","2345078736","3422843424","3548374830","2560297740","1224516150","349367436","45046719"],["2704156","48674808","399627228","1978736760","6580546830","15484030848","26431348944","32978129184","29847357540","19109710620","8215325118","2129223564","251595969"],["10400600","202811700","1817192832","9905323428","36645428820","97161794730","189936558240","277162821936","301896418824","242419757580","139482383040","54455105250","12929165886","1409933619"],["40116600","842448600","8180071900","48674808000","198284942856","584949505140","1288652314950","2153597228640","2743499118360","2651111511048","1912859528580","999301605120","357302804040","78265638990","7923848253"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","37","305","2641","23525","213445","1961825","18205345","170195525","1600472677","15122515985","143457011569","1365435096485","13033485491077"],"nonn":true,"definitions":[{"position":0,"source":"derived:reversal","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["-6","-15","-8"],["4","9","4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-2","-6","-4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":183885,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-1, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":false,"value":"-1"},"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-1, 0]","region_exact":null,"edges":["e-6592992e1b6851ae"]},{"id":183886,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-6592992e1b6851ae"]},{"id":183887,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-6592992e1b6851ae"]}],"region_exact":{},"incoming_edges":[{"id":"e-6592992e1b6851ae","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"reversal\",\"schedule\":{\"first_index\":1,\"first_degree\":0,\"step\":1}}"},"operation_detail":"reversal, shift 0","conclusion_id":"n-a882d0a2ea16bc46","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":"reversed","lean_declarations":["RealRooted.X_pow_mul_reverse_eq_zero_or_splits","RealRooted.reflect_eq_zero_or_splits"],"certificate_hash":"92a05b50c96b76b718009160ac774e2a8e90f15d3e58c2ab3e32e01ef17139ae","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":false,"value":"-1"},"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-0444a4f352052985","role":"sequence","name":"basis_map_A007318 of T(n,k) = binomial(n,k)*binomial(n+k,k), 0 <= k <= n, triangle read by rows.","uses":{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":false,"value":"-2"},"upper":{"closed":false,"value":"-1"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-2, -1)"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-1b667780249660ca","kind":"candidate_equal","node_a_id":"n-3f2e898dea3fb1c5","node_b_id":"n-a882d0a2ea16bc46","fingerprint":null,"oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"reason":"no all-index equality certificate was found","rows_compared":15},"other_id":"n-3f2e898dea3fb1c5","other":{"id":"n-3f2e898dea3fb1c5","name":"mobius_x_over_x_plus_one of Triangle read by rows: T(n,k) is the number of lattice paths from (0,0) to (n,n) using steps E=(1,0), N=(0,1) and D=(1,1) (i.e., bilateral Schroeder paths), having k D=(1,1) steps.","first_index":1,"fingerprint":"f6-c277ed2a39f7c92c","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","3"],["6","18","13"]],"card_rows":[["1"],["2","3"],["6","18","13"],["20","90","132","63"]],"row_sums_preview":["1","5","37","305","2641","23525","213445","1961825"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-78ba05b4a3dd925d","steps":["x \u21a6 x/(x+1)"],"edges":["e-22fed0d4659f5966"],"text":"A104684 \u2192 x \u21a6 x/(x+1)"},"headline":"A104684 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-176362b653bfc94e","kind":"fingerprint_related","node_a_id":"n-a882d0a2ea16bc46","node_b_id":"n-a0d580d2a8ee89f3","fingerprint":"f6-c277ed2a39f7c92c","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-c277ed2a39f7c92c"},"other_id":"n-a0d580d2a8ee89f3","other":{"id":"n-a0d580d2a8ee89f3","name":"mobius_x_over_x_plus_one of basis_map_A007318 of Triangle T(n, k) of numbers of square lattice walks that start and end at origin after 2*n steps and contain exactly k steps to the east, possibly touching origin at intermediate stages.","first_index":0,"fingerprint":"f6-c277ed2a39f7c92c","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["4","6"],["36","108","78"]],"card_rows":[["1"],["4","6"],["36","108","78"],["400","1800","2640","1260"]],"row_sums_preview":["1","10","222","6100","184870","5928300","197223180","6732983400"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-fdc03fad975441d0","steps":["basis A007318","x \u21a6 x/(x+1)"],"edges":["e-2cd9cd27e716475b","e-821b6024dcf49c9c"],"text":"A069466 \u2192 basis A007318 \u2192 x \u21a6 x/(x+1)"},"headline":"A069466 \u2192 basis A007318 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-1e745d1d7eb25533","kind":"fingerprint_related","node_a_id":"n-0444a4f352052985","node_b_id":"n-a882d0a2ea16bc46","fingerprint":"f6-c277ed2a39f7c92c","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-c277ed2a39f7c92c"},"other_id":"n-0444a4f352052985","other":{"id":"n-0444a4f352052985","name":"basis_map_A007318 of T(n,k) = binomial(n,k)*binomial(n+k,k), 0 <= k <= n, triangle read by rows.","first_index":1,"fingerprint":"f6-c277ed2a39f7c92c","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["3","2"],["13","18","6"]],"card_rows":[["1"],["3","2"],["13","18","6"],["63","132","90","20"]],"row_sums_preview":["1","5","37","305","2641","23525","213445","1961825"],"regions":["(-inf, 0]","(-2, -1)"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4689fb1386d440ad","steps":["x \u21a6 x+1"],"edges":["e-437a2dd956a9df17"],"text":"A063007 \u2192 x \u21a6 x+1"},"headline":"A063007 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-eef2c92adea0e1bc","kind":"fingerprint_related","node_a_id":"n-c3e7c36a90208dd3","node_b_id":"n-a882d0a2ea16bc46","fingerprint":"f6-c277ed2a39f7c92c","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-c277ed2a39f7c92c"},"other_id":"n-c3e7c36a90208dd3","other":{"id":"n-c3e7c36a90208dd3","name":"mobius_x_over_x_plus_one of basis_map_A007318 of Square the entries of Pascal's triangle.","first_index":0,"fingerprint":"f6-c277ed2a39f7c92c","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","3"],["6","18","13"]],"card_rows":[["1"],["2","3"],["6","18","13"],["20","90","132","63"]],"row_sums_preview":["1","5","37","305","2641","23525","213445","1961825"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-b13531ad79e7c629","steps":["x \u21a6 x+1","x \u21a6 x/(x+1)"],"edges":["e-041c07f9bf4043d7","e-8ff3c2ff69ba28ab"],"text":"A008459 \u2192 x \u21a6 x+1 \u2192 x \u21a6 x/(x+1)"},"headline":"A008459 \u2192 x \u21a6 x+1 \u2192 x \u21a6 x/(x+1)","chain":null}}],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:     1;\n   2:     2,      3;\n   3:     6,     18,      13;\n   4:    20,     90,     132,       63;\n   5:    70,    420,     930,      900,      321;\n   6:   252,   1890,    5600,     8190,     5910,     1683;\n   7:   924,   8316,   30870,    60480,    65940,    37926,     8989;\n   8:  3432,  36036,  160776,   395010,   577080,   501228,   239624,    48639;\n   9: 12870, 154440,  804804,  2378376,  4358970,  5072760,  3660300,  1497096,  265729;\n  10: 48620, 656370, 3912480, 13513500, 29801772, 43513470, 42060480, 25951860, 9274410, 1462563;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = (4*n-6+(9*n-15)*x+(4*n-8)*x^2)*P(n-1,x) - (2*x+6*x^2+4*x^3)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((4*n - 6 + (9*n - 15)*x + (4*n - 8)*x^2)*P[n - 1, x] - (2*x + 6*x^2 + 4*x^3)*D[P[n - 1, x], x])/(n - 1)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((4*n - 6 + (9*n - 15)*x + (4*n - 8)*x^2)*v[i - 1] - (2*x + 6*x^2 + 4*x^3)*deriv(v[i - 1], x))/(n - 1)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-4689fb1386d440ad","steps":["x \u21a6 x+1","reverse"],"edges":["e-437a2dd956a9df17","e-6592992e1b6851ae"],"text":"A063007 \u2192 x \u21a6 x+1 \u2192 reverse"},"headline":"A063007 \u2192 x \u21a6 x+1 \u2192 reverse","chain":null}