{"id":"n-036d2f1a653e875d","revision":0,"first_index":0,"fingerprint":"f6-2f8272d81c3bae1a","name":"derivative of Triangle read by rows: T(n,0)=1, T(n,n)=2, T(n,m) = T(n-1,m-1) + Sum_{k=0..m} T(n-1-k,m-k).","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":[[],["2"],["4","4"],["6","18","6"],["8","40","48","8"],["10","70","150","100","10"],["12","108","336","420","180","12"],["14","154","630","1176","980","294","14"],["16","208","1056","2640","3360","2016","448","16"],["18","270","1638","5148","8910","8316","3780","648","18"],["20","340","2400","9100","20020","25740","18480","6600","900","20"],["22","418","3366","14960","40040","66066","66066","37752","10890","1210","22"],["24","504","4560","23256","73440","148512","192192","154440","72072","17160","1584","24"],["26","598","6006","34580","125970","302328","482664","505648","334620","130130","26026","2028","26"],["28","700","7728","49588","204820","569772","1085280","1410864","1225224","680680","224224","38220","2548","28"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","2","8","30","104","340","1068","3262","9760","28746","83620","240812","687768","1950650","5499704"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["2"]],"recurrence":{"denominator":{"coeffs":[["3"],["-5"],["2"]]},"terms":[{"coeff":{"coeffs":[["0","0"],["-3","-11"],["2","6"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","0"],["0","-1","-4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":172984,"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-05789d3edae5e2c9"]},{"id":172985,"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-05789d3edae5e2c9"]},{"id":172986,"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-05789d3edae5e2c9"]},{"id":172987,"position":3,"claim":"interlacing","from_index":2,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2","region_exact":null,"edges":["e-05789d3edae5e2c9"]}],"region_exact":{},"incoming_edges":[{"id":"e-05789d3edae5e2c9","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"0\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-036d2f1a653e875d","premise_count":1,"theorem":"Laguerre's theorem on differential operators: if phi is a real polynomial with only real roots, then phi(D) maps every real-rooted polynomial to a real-rooted polynomial or zero (write phi(D) = C D^j prod (1 + c_i D); D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"6be4ed6418c0c73a337c41822da26cd30e98d9c39e94865d7b95aae6ddc11376","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-2496e2f555756362","role":"sequence","name":"Triangle read by rows: T(n,0)=1, T(n,n)=2, T(n,m) = T(n-1,m-1) + Sum_{k=0..m} T(n-1-k,m-k).","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-1ec28a3de8472185","kind":"fingerprint_related","node_a_id":"n-ff36538314ab71f5","node_b_id":"n-036d2f1a653e875d","fingerprint":"f6-2f8272d81c3bae1a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2f8272d81c3bae1a"},"other_id":"n-ff36538314ab71f5","other":{"id":"n-ff36538314ab71f5","name":"basis_map_A013611 of Triangle read by rows: T(n,k) = binomial(2n+1, 2k+1).","first_index":0,"fingerprint":"f6-2f8272d81c3bae1a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["4","4"],["16","48","16"]],"card_rows":[["1"],["4","4"],["16","48","16"],["64","320","384","64"]],"row_sums_preview":["1","8","80","832","8704","91136","954368","9994240"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c30cb06ad75c5e21","steps":["basis A013611"],"edges":["e-36be6b9e5c42de00"],"text":"A103327 \u2192 basis A013611"},"headline":"A103327 \u2192 basis A013611","chain":null}},{"id":"l-312a713ea57ce81e","kind":"fingerprint_related","node_a_id":"n-036d2f1a653e875d","node_b_id":"n-d241d6f524076063","fingerprint":"f6-2f8272d81c3bae1a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2f8272d81c3bae1a"},"other_id":"n-d241d6f524076063","other":{"id":"n-d241d6f524076063","name":"derivative of basis_map_A013611 of Triangle read by rows: T(n, k) = binomial(2n, 2k).","first_index":1,"fingerprint":"f6-2f8272d81c3bae1a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["4"],["32","32"]],"card_rows":[["4"],["32","32"],["192","576","192"],["1024","5120","6144","1024"]],"row_sums_preview":["0","4","64","960","13312","174080","2187264","26722304"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-f7935c36a3c0ec8c","steps":["basis A013611","D"],"edges":["e-11143fa0a471b62a","e-069228bccf40c3b2"],"text":"A086645 \u2192 basis A013611 \u2192 D"},"headline":"A086645 \u2192 basis A013611 \u2192 D","chain":null}},{"id":"l-6e51e2bf97530085","kind":"fingerprint_related","node_a_id":"n-0eba401b117f16a9","node_b_id":"n-036d2f1a653e875d","fingerprint":"f6-2f8272d81c3bae1a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2f8272d81c3bae1a"},"other_id":"n-0eba401b117f16a9","other":{"id":"n-0eba401b117f16a9","name":"Triangle read by rows: T(n, k) = binomial(n + k, 2*k).","first_index":1,"fingerprint":"f6-2f8272d81c3bae1a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","1"],["1","3","1"]],"card_rows":[["1"],["1","1"],["1","3","1"],["1","6","5","1"]],"row_sums_preview":["1","2","5","13","34","89","233","610"],"regions":["(-inf, +inf)","(-inf, 0)","(-4, 0)"],"region_exact":{},"oeis_ids":["A085478"],"derivation":{"root":"n-0eba401b117f16a9","steps":[],"edges":[],"text":"A085478"},"headline":"A085478: Triangle read by rows: T(n, k) = binomial(n + k, 2*k).","chain":null}},{"id":"l-b8c66cdc93abb52f","kind":"fingerprint_related","node_a_id":"n-01fe968f3e05a927","node_b_id":"n-036d2f1a653e875d","fingerprint":"f6-2f8272d81c3bae1a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2f8272d81c3bae1a"},"other_id":"n-01fe968f3e05a927","other":{"id":"n-01fe968f3e05a927","name":"Triangle T(n,k), read by rows given by [1,0,1,0,0,0,0,0,0,...] DELTA [0,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.","first_index":0,"fingerprint":"f6-2f8272d81c3bae1a","polytool_family":null,"degree_pattern":"irregular","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1"],["1","1"]],"card_rows":[["1"],["1"],["1","1"],["1","3","1"]],"row_sums_preview":["1","1","2","5","13","34","89","233"],"regions":["(-inf, +inf)","(-inf, 0)","(-4, 0)","(-inf, 0]"],"region_exact":{},"oeis_ids":["A165253"],"derivation":{"root":"n-01fe968f3e05a927","steps":[],"edges":[],"text":"A165253"},"headline":"A165253: Triangle T(n,k), read by rows given by [1,0,1,0,0,0,0,0,0,...] DELTA [0,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.","chain":null}},{"id":"l-c29882b23849b2ad","kind":"fingerprint_related","node_a_id":"n-4c232e962100ae75","node_b_id":"n-036d2f1a653e875d","fingerprint":"f6-2f8272d81c3bae1a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2f8272d81c3bae1a"},"other_id":"n-4c232e962100ae75","other":{"id":"n-4c232e962100ae75","name":"reversal of Triangular array binomial(2*n-k, k), k=0..n, n >= 0.","first_index":0,"fingerprint":"f6-2f8272d81c3bae1a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","1"],["1","3","1"]],"card_rows":[["1"],["1","1"],["1","3","1"],["1","6","5","1"]],"row_sums_preview":["1","2","5","13","34","89","233","610"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-6758fe7729845be8","steps":["reverse"],"edges":["e-597fd348d5cdfa65"],"text":"A054142 \u2192 reverse"},"headline":"A054142 \u2192 reverse","chain":null}},{"id":"l-cfbdf858aef83be2","kind":"fingerprint_related","node_a_id":"n-d8cfb1e39c5d56b0","node_b_id":"n-036d2f1a653e875d","fingerprint":"f6-2f8272d81c3bae1a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2f8272d81c3bae1a"},"other_id":"n-d8cfb1e39c5d56b0","other":{"id":"n-d8cfb1e39c5d56b0","name":"reversal of Triangle read by rows: T(n, k) = binomial(n + k, 2*k).","first_index":1,"fingerprint":"f6-2f8272d81c3bae1a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","1"],["1","3","1"]],"card_rows":[["1"],["1","1"],["1","3","1"],["1","5","6","1"]],"row_sums_preview":["1","2","5","13","34","89","233","610"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-0eba401b117f16a9","steps":["reverse"],"edges":["e-6116c33129f68e6c"],"text":"A085478 \u2192 reverse"},"headline":"A085478 \u2192 reverse","chain":null}},{"id":"l-d4954c3cfd1008a7","kind":"fingerprint_related","node_a_id":"n-3af57aaf5c129e4b","node_b_id":"n-036d2f1a653e875d","fingerprint":"f6-2f8272d81c3bae1a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2f8272d81c3bae1a"},"other_id":"n-3af57aaf5c129e4b","other":{"id":"n-3af57aaf5c129e4b","name":"Triangle T(n,k), 0 <= k <= n, read by rows given by [0, 1, 0, 0, 0, 0, ...] DELTA [1, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.","first_index":0,"fingerprint":"f6-2f8272d81c3bae1a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["0","1"],["0","1","1"]],"card_rows":[["1"],["0","1"],["0","1","1"],["0","1","3","1"]],"row_sums_preview":["1","1","2","5","13","34","89","233"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":["A121314"],"derivation":{"root":"n-3af57aaf5c129e4b","steps":[],"edges":[],"text":"A121314"},"headline":"A121314: Triangle T(n,k), 0 <= k <= n, read by rows given by [0, 1, 0, 0, 0, 0, ...] DELTA [1, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.","chain":null}},{"id":"l-dee1ee09585bb554","kind":"fingerprint_related","node_a_id":"n-6758fe7729845be8","node_b_id":"n-036d2f1a653e875d","fingerprint":"f6-2f8272d81c3bae1a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2f8272d81c3bae1a"},"other_id":"n-6758fe7729845be8","other":{"id":"n-6758fe7729845be8","name":"Triangular array binomial(2*n-k, k), k=0..n, n >= 0.","first_index":0,"fingerprint":"f6-2f8272d81c3bae1a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","1"],["1","3","1"]],"card_rows":[["1"],["1","1"],["1","3","1"],["1","5","6","1"]],"row_sums_preview":["1","2","5","13","34","89","233","610"],"regions":["(-inf, +inf)","(-inf, 0)","(-inf, 0]"],"region_exact":{},"oeis_ids":["A054142"],"derivation":{"root":"n-6758fe7729845be8","steps":[],"edges":[],"text":"A054142"},"headline":"A054142: Triangular array binomial(2*n-k, k), k=0..n, n >= 0.","chain":null}},{"id":"l-e28a245194f38334","kind":"fingerprint_related","node_a_id":"n-036d2f1a653e875d","node_b_id":"n-6439dbf6afbb8021","fingerprint":"f6-2f8272d81c3bae1a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2f8272d81c3bae1a"},"other_id":"n-6439dbf6afbb8021","other":{"id":"n-6439dbf6afbb8021","name":"reversal of basis_map_A013611 of Triangle read by rows: T(n,k) = binomial(2n+1, 2k+1).","first_index":0,"fingerprint":"f6-2f8272d81c3bae1a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["4","4"],["16","48","16"]],"card_rows":[["1"],["4","4"],["16","48","16"],["64","384","320","64"]],"row_sums_preview":["1","8","80","832","8704","91136","954368","9994240"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c30cb06ad75c5e21","steps":["basis A013611","reverse"],"edges":["e-36be6b9e5c42de00","e-085a3589b4ff240c"],"text":"A103327 \u2192 basis A013611 \u2192 reverse"},"headline":"A103327 \u2192 basis A013611 \u2192 reverse","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:  0;\n  1:  2;\n  2:  4,   4;\n  3:  6,  18,    6;\n  4:  8,  40,   48,    8;\n  5: 10,  70,  150,  100,   10;\n  6: 12, 108,  336,  420,  180,   12;\n  7: 14, 154,  630, 1176,  980,  294,   14;\n  8: 16, 208, 1056, 2640, 3360, 2016,  448,  16;\n  9: 18, 270, 1638, 5148, 8910, 8316, 3780, 648, 18;\n","recurrence_sentence":"The row polynomials satisfy (2*n^2-5*n+3)*P(n,x) = (2*n^2-3*n+(6*n^2-11*n)*x)*P(n-1,x) - (n*x+4*n*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n^2 - 3*n + (6*n^2 - 11*n)*x)*P[n - 1, x] - (n*x + 4*n*x^2)*D[P[n - 1, x], x])/(2*n^2 - 5*n + 3)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = ((2*n^2 - 3*n + (6*n^2 - 11*n)*x)*v[i - 1] - (n*x + 4*n*x^2)*deriv(v[i - 1], x))/(2*n^2 - 5*n + 3)); 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-2496e2f555756362","steps":["D"],"edges":["e-05789d3edae5e2c9"],"text":"A084534 \u2192 D"},"headline":"A084534 \u2192 D","chain":null}
