{"id":"n-4ca874876022e136","revision":1,"first_index":0,"fingerprint":"f6-5f98bd4aea48ed7d","name":"Triangle read by rows, T(n, k) = binomial(n, k)*binomial(n + 2, k).","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"],["1","3"],["1","8","6"],["1","15","30","10"],["1","24","90","80","15"],["1","35","210","350","175","21"],["1","48","420","1120","1050","336","28"],["1","63","756","2940","4410","2646","588","36"],["1","80","1260","6720","14700","14112","5880","960","45"],["1","99","1980","13860","41580","58212","38808","11880","1485","55"],["1","120","2970","26400","103950","199584","194040","95040","22275","2200","66"],["1","143","4290","47190","235950","594594","792792","566280","212355","39325","3146","78"],["1","168","6006","80080","495495","1585584","2774772","2718144","1486485","440440","66066","4368","91"],["1","195","8190","130130","975975","3864861","8588580","11042460","8281845","3578575","858858","106470","5915","105"],["1","224","10920","203840","1821820","8744736","24048024","39262080","38648610","22902880","8016008","1589952","165620","7840","120"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","4","15","56","210","792","3003","11440","43758","167960","646646","2496144","9657700","37442160","145422675"],"nonn":true,"definitions":[{"position":0,"source":"oeis_catalog:A062196","source_kind":"oeis_catalog","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["0"],["2"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","2"],["2","4"],["1","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","2","-2"],["0","2","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":158794,"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":false,"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-32e7e43d5a9debda"]},{"id":158795,"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-32e7e43d5a9debda"]},{"id":158796,"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-32e7e43d5a9debda"]},{"id":158797,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"previous_interlaces_next"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m >= 1","region_exact":null,"edges":["e-32e7e43d5a9debda"]}],"region_exact":{},"incoming_edges":[{"id":"e-32e7e43d5a9debda","operation":{"operation":"theorem_check","theorem":"liu_wang"},"operation_detail":"liu_wang","conclusion_id":"n-4ca874876022e136","premise_count":0,"theorem":"L. L. Liu and Y. Wang, A unified approach to polynomial sequences with only real zeros, Adv. in Appl. Math. 38 (2007) 542-560, Theorem 2.1 with g = f' (as in their Corollary 2.4), with nonnegativity of the coefficients replaced by an exact induction keeping every root in (-inf, 0)","lean_route":null,"lean_declarations":[],"certificate_hash":"a4d53391444213039ae5b5927b1e3dc1adf4666981e82148f207edb8ffc16007","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":false,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"previous_interlaces_next"}],"premises":[]}],"used_by":[{"id":"e-039745f356dca7e2","operation":"obreschkoff_combination","operation_detail":null,"conclusion_id":"n-137a24c9ba6939cb","conclusion_name":"obreschkoff_combination of Triangle read by rows, T(n, k) = binomial(n, k)*binomial(n + 2, k)."},{"id":"e-039745f356dca7e2","operation":"obreschkoff_combination","operation_detail":null,"conclusion_id":"n-137a24c9ba6939cb","conclusion_name":"obreschkoff_combination of Triangle read by rows, T(n, k) = binomial(n, k)*binomial(n + 2, k)."},{"id":"e-05dbfee114f87ac9","operation":"preserver_image","operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-9e1f147e35ceaa1b","conclusion_name":"theta_plus_one of Triangle read by rows, T(n, k) = binomial(n, k)*binomial(n + 2, k)."},{"id":"e-2305188708ae9104","operation":"preserver_image","operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-cff3d698fea91c3e","conclusion_name":"one_plus_derivative of Triangle read by rows, T(n, k) = binomial(n, k)*binomial(n + 2, k)."},{"id":"e-2ae8453bca041506","operation":"preserver_image","operation_detail":"affine, shift 0","conclusion_id":"n-5aa52e3acf29a627","conclusion_name":"Triangle read by rows, related to clusters of type D."},{"id":"e-6f314a2e7485472a","operation":"preserver_image","operation_detail":"affine, shift 0","conclusion_id":"n-5aa52e3acf29a627","conclusion_name":"Triangle read by rows, related to clusters of type D."},{"id":"e-7729d1773ce70fbd","operation":"preserver_image","operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-82cc9c378f27e13e","conclusion_name":"derivative of Triangle read by rows, T(n, k) = binomial(n, k)*binomial(n + 2, k)."},{"id":"e-a3706a3507b0ac5d","operation":"preserver_image","operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-2e37cda1e7c758ee","conclusion_name":"theta_plus_one_squared of Triangle read by rows, T(n, k) = binomial(n, k)*binomial(n + 2, k)."},{"id":"e-b6d3c37eae668953","operation":"preserver_image","operation_detail":"reversal, shift 0","conclusion_id":"n-3f65f4dc3d5d123d","conclusion_name":"reversal of Triangle read by rows, T(n, k) = binomial(n, k)*binomial(n + 2, k)."},{"id":"e-f6a7bd41834fc468","operation":"preserver_image","operation_detail":"mobius, shift 0","conclusion_id":"n-fc24cdc1c25a5a29","conclusion_name":"mobius_x_over_x_plus_one of Triangle read by rows, T(n, k) = binomial(n, k)*binomial(n + 2, k)."}],"used_by_total":10,"links":[{"id":"l-8534329e75f409e7","kind":"fingerprint_related","node_a_id":"n-4ca874876022e136","node_b_id":"n-3f65f4dc3d5d123d","fingerprint":"f6-5f98bd4aea48ed7d","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-5f98bd4aea48ed7d"},"other_id":"n-3f65f4dc3d5d123d","other":{"id":"n-3f65f4dc3d5d123d","name":"reversal of Triangle read by rows, T(n, k) = binomial(n, k)*binomial(n + 2, k).","first_index":0,"fingerprint":"f6-5f98bd4aea48ed7d","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["3","1"],["6","8","1"]],"card_rows":[["1"],["3","1"],["6","8","1"],["10","30","15","1"]],"row_sums_preview":["1","4","15","56","210","792","3003","11440"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4ca874876022e136","steps":["reverse"],"edges":["e-b6d3c37eae668953"],"text":"A062196 \u2192 reverse"},"headline":"A062196 \u2192 reverse","chain":null}},{"id":"l-a8b098feca228cbf","kind":"oeis_match","node_a_id":"n-4ca874876022e136","node_b_id":null,"fingerprint":null,"oeis_id":"A001791","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":"A001791","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-b4b47c8d1457fc84","kind":"oeis_match","node_a_id":"n-4ca874876022e136","node_b_id":null,"fingerprint":null,"oeis_id":"A062196","in_catalog":1,"target":"triangle","row_alignment":"tabl","rows_matched":10,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":10,"target":"triangle","terms_compared":55},"in_catalog":true,"oeis_id":"A062196","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A062196","relation":"identification"},{"oeis_id":"A001791","relation":"match_row_sums"},{"oeis_id":"A062196","relation":"match_triangle"}],"bfiles":[{"node_id":"n-4ca874876022e136","oeis_id":"A062196","target":"triangle","oeis_offset":0,"first_row":0,"rows_used":44,"terms_matched":55,"oeis_terms_agreeing":55,"oeis_terms_available":55,"term_count":990,"raw_size":16669}],"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: 1,  3;\n  2: 1,  8,    6;\n  3: 1, 15,   30,    10;\n  4: 1, 24,   90,    80,    15;\n  5: 1, 35,  210,   350,   175,    21;\n  6: 1, 48,  420,  1120,  1050,   336,    28;\n  7: 1, 63,  756,  2940,  4410,  2646,   588,    36;\n  8: 1, 80, 1260,  6720, 14700, 14112,  5880,   960,   45;\n  9: 1, 99, 1980, 13860, 41580, 58212, 38808, 11880, 1485, 55;\n","recurrence_sentence":"The row polynomials satisfy (n^2+2*n)*P(n,x) = (n^2+2*n+(3*n^2+4*n+2)*x)*P(n-1,x) + ((2*n+2)*x-(2*n+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[Together[((n^2 + 2*n + (3*n^2 + 4*n + 2)*x)*P[n - 1, x] + ((2*n + 2)*x - (2*n + 2)*x^2)*D[P[n - 1, x], x])/(n^2 + 2*n)]];\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] = ((n^2 + 2*n + (3*n^2 + 4*n + 2)*x)*v[i - 1] + ((2*n + 2)*x - (2*n + 2)*x^2)*deriv(v[i - 1], x))/(n^2 + 2*n)); 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}},{"oeis_id":"A062196","target":"triangle","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0: 1;\n  1: 1,  3;\n  2: 1,  8,    6;\n  3: 1, 15,   30,    10;\n  4: 1, 24,   90,    80,    15;\n  5: 1, 35,  210,   350,   175,    21;\n  6: 1, 48,  420,  1120,  1050,   336,    28;\n  7: 1, 63,  756,  2940,  4410,  2646,   588,    36;\n  8: 1, 80, 1260,  6720, 14700, 14112,  5880,   960,   45;\n  9: 1, 99, 1980, 13860, 41580, 58212, 38808, 11880, 1485, 55;\n","recurrence_sentence":"The row polynomials satisfy (n^2+2*n)*P(n,x) = (n^2+2*n+(3*n^2+4*n+2)*x)*P(n-1,x) + ((2*n+2)*x-(2*n+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[Together[((n^2 + 2*n + (3*n^2 + 4*n + 2)*x)*P[n - 1, x] + ((2*n + 2)*x - (2*n + 2)*x^2)*D[P[n - 1, x], x])/(n^2 + 2*n)]];\nTable[PadRight[CoefficientList[P[n, x], x], n + 1], {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] = ((n^2 + 2*n + (3*n^2 + 4*n + 2)*x)*v[i - 1] + ((2*n + 2)*x - (2*n + 2)*x^2)*deriv(v[i - 1], x))/(n^2 + 2*n)); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); 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":true}}],"derivation":{"root":"n-4ca874876022e136","steps":[],"edges":[],"text":"A062196"},"headline":"A062196: Triangle read by rows, T(n, k) = binomial(n, k)*binomial(n + 2, k).","chain":null}
