{"id":"n-037f885e93f00c5b","revision":0,"first_index":0,"fingerprint":"f6-3d0fbbfa79b3fd61","name":"reversal of Triangle read by rows, T(n,k) = (k+1)*(n+1)!*(n+k)!/((k+1)!^2*(n-k)!) with n >= 0 and 0 <= k <= n.","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","2"],["12","18","6"],["120","240","144","24"],["1680","4200","3600","1200","120"],["30240","90720","100800","50400","10800","720"],["665280","2328480","3175200","2116800","705600","105840","5040"],["17297280","69189120","111767040","93139200","42336000","10160640","1128960","40320"],["518918400","2335132800","4358914560","4358914560","2514758400","838252800","152409600","13063680","362880"],["17643225600","88216128000","186810624000","217945728000","152562009600","65383718400","16765056000","2395008000","163296000","3628800"],["670442572800","3687434150400","8733396672000","11644528896000","9589612032000","5034546316800","1678182105600","342486144000","39517632000","2195424000","39916800"],["28158588057600","168951528345600","442492098048000","663738147072000","628804560384000","391256170905600","161105482137600","43153254144000","7192209024000","684972288000","31614105600","479001600"],["1295295050649600","8419417829222400","24160068553420800","40266780922368000","43142979559680000","31062945282969600","15258990665318400","5086330221772800","1121984607744000","155831195520000","12466495641600","485707622400","6227020800"],["64764752532480000","453353267727360000","1414462195309363200","2593180691400499200","3100542131022336000","2536807198109184000","1449604113205248000","579841645282099200","160219401985843200","29670259627008000","3490618779648000","237996734976000","7933224499200","87178291200"],["3497296636753920000","26229724775654400000","88403887206835200000","176807774413670400000","233386262226044928000","213937407040541184000","139524395896005120000","65232185094236160000","21744061698078720000","5073614396218368000","801097009929216000","80918889891840000","4759934699520000","137305808640000","1307674368000"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","4","36","528","10800","283680","9102240","345058560","15090727680","747888422400","41422381862400","2535569103513600","169983582318950400","12386182292118835200","974723523832041984000"],"nonn":true,"definitions":[{"position":0,"source":"derived:reversal","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-2","-1"],["4","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-2","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":190874,"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-620301680ebc6d01"]},{"id":190875,"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-620301680ebc6d01"]},{"id":190876,"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-620301680ebc6d01"]}],"region_exact":{},"incoming_edges":[{"id":"e-620301680ebc6d01","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"reversal\",\"schedule\":{\"first_index\":0,\"first_degree\":0,\"step\":1}}"},"operation_detail":"reversal, shift 0","conclusion_id":"n-037f885e93f00c5b","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":"1a79294b70033b257ee3fbe63dd9d8f971c945fb152a8418799a5e6570525661","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-694017a14c5c1f2e","role":"sequence","name":"Triangle read by rows, T(n,k) = (k+1)*(n+1)!*(n+k)!/((k+1)!^2*(n-k)!) with n >= 0 and 0 <= k <= n.","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-5575bc02fa78ba52","kind":"fingerprint_related","node_a_id":"n-694017a14c5c1f2e","node_b_id":"n-037f885e93f00c5b","fingerprint":"f6-3d0fbbfa79b3fd61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-3d0fbbfa79b3fd61"},"other_id":"n-694017a14c5c1f2e","other":{"id":"n-694017a14c5c1f2e","name":"Triangle read by rows, T(n,k) = (k+1)*(n+1)!*(n+k)!/((k+1)!^2*(n-k)!) with n >= 0 and 0 <= k <= n.","first_index":0,"fingerprint":"f6-3d0fbbfa79b3fd61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","2"],["6","18","12"]],"card_rows":[["1"],["2","2"],["6","18","12"],["24","144","240","120"]],"row_sums_preview":["1","4","36","528","10800","283680","9102240","345058560"],"regions":["(-inf, 0)"],"region_exact":{},"oeis_ids":["A253284"],"derivation":{"root":"n-694017a14c5c1f2e","steps":[],"edges":[],"text":"A253284"},"headline":"A253284: Triangle read by rows, T(n,k) = (k+1)*(n+1)!*(n+k)!/((k+1)!^2*(n-k)!) with n >= 0 and 0 <= k <= n.","chain":null}},{"id":"l-65640ef8aa949609","kind":"fingerprint_related","node_a_id":"n-df818498d48ea142","node_b_id":"n-037f885e93f00c5b","fingerprint":"f6-3d0fbbfa79b3fd61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-3d0fbbfa79b3fd61"},"other_id":"n-df818498d48ea142","other":{"id":"n-df818498d48ea142","name":"reversal of Triangle read by rows: T(n,k) = C(n+k,n)*C(n,k)/(k+1), for n >= 0, k = 0..n.","first_index":1,"fingerprint":"f6-3d0fbbfa79b3fd61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","1"],["2","3","1"]],"card_rows":[["1"],["1","1"],["2","3","1"],["5","10","6","1"]],"row_sums_preview":["1","2","6","22","90","394","1806","8558"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-be392fc9b98fb771","steps":["reverse"],"edges":["e-d48739f504014efc"],"text":"A088617 \u2192 reverse"},"headline":"A088617 \u2192 reverse","chain":null}},{"id":"l-c0c3b0d946c54ceb","kind":"fingerprint_related","node_a_id":"n-be392fc9b98fb771","node_b_id":"n-037f885e93f00c5b","fingerprint":"f6-3d0fbbfa79b3fd61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-3d0fbbfa79b3fd61"},"other_id":"n-be392fc9b98fb771","other":{"id":"n-be392fc9b98fb771","name":"Triangle read by rows: T(n,k) = C(n+k,n)*C(n,k)/(k+1), for n >= 0, k = 0..n.","first_index":1,"fingerprint":"f6-3d0fbbfa79b3fd61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","1"],["1","3","2"]],"card_rows":[["1"],["1","1"],["1","3","2"],["1","6","10","5"]],"row_sums_preview":["1","2","6","22","90","394","1806","8558"],"regions":["(-inf, 0)"],"region_exact":{},"oeis_ids":["A088617"],"derivation":{"root":"n-be392fc9b98fb771","steps":[],"edges":[],"text":"A088617"},"headline":"A088617: Triangle read by rows: T(n,k) = C(n+k,n)*C(n,k)/(k+1), for n >= 0, k = 0..n.","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:           2,           2;\n  2:          12,          18,            6;\n  3:         120,         240,          144,           24;\n  4:        1680,        4200,         3600,         1200,          120;\n  5:       30240,       90720,       100800,        50400,        10800,         720;\n  6:      665280,     2328480,      3175200,      2116800,       705600,      105840,        5040;\n  7:    17297280,    69189120,    111767040,     93139200,     42336000,    10160640,     1128960,      40320;\n  8:   518918400,  2335132800,   4358914560,   4358914560,   2514758400,   838252800,   152409600,   13063680,    362880;\n  9: 17643225600, 88216128000, 186810624000, 217945728000, 152562009600, 65383718400, 16765056000, 2395008000, 163296000, 3628800;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (4*n-2+(3*n-1)*x)*P(n-1,x) - (2*x+2*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(4*n - 2 + (3*n - 1)*x)*P[n - 1, x] - (2*x + 2*x^2)*D[P[n - 1, x], x]];\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] = (4*n - 2 + (3*n - 1)*x)*v[i - 1] - (2*x + 2*x^2)*deriv(v[i - 1], x)); 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-694017a14c5c1f2e","steps":["reverse"],"edges":["e-620301680ebc6d01"],"text":"A253284 \u2192 reverse"},"headline":"A253284 \u2192 reverse","chain":null}