{"id":"n-daf15103e37c2e90","revision":0,"first_index":1,"fingerprint":"f6-e7a207c59c800cef","name":"shift_x_plus_one of derivative of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by upward antidiagonals.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":-3,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[[],[],["1"],["2"],["5","2"],["10","6"],["20","18","3"],["38","44","12"],["71","102","42","4"],["130","222","120","20"],["235","466","315","80","5"],["420","948","768","260","30"],["744","1884","1782","760","135","6"],["1308","3672","3972","2044","490","42"],["2285","7044","8580","5180","1575","210","7"]],"degrees":[-1,-1,0,0,1,1,2,2,3,3,4,4,5,5,6],"row_sums":["0","0","1","2","7","16","41","94","219","492","1101","2426","5311","11528","24881"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["0"],["0"],["1"]],"recurrence":{"denominator":{"coeffs":[["-3"],["1"]]},"terms":[{"coeff":{"coeffs":[["-10","-8"],["3","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-5","-9","-4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":171146,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, -1]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"-1"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, -1]","region_exact":null,"edges":["e-8b7b972ac21d1c05"]},{"id":171147,"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-8b7b972ac21d1c05","e-f0833858828f3c1d"]},{"id":171148,"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-8b7b972ac21d1c05","e-f0833858828f3c1d"]},{"id":171149,"position":3,"claim":"real_rooted","from_index":1,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-f0833858828f3c1d"]}],"region_exact":{},"incoming_edges":[{"id":"e-8b7b972ac21d1c05","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-daf15103e37c2e90","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":"b3c6df6139cba64b162512e2de6a40573baf87958c14ed523866d76bbcc5d74a","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"-1"}}}],"premises":[{"position":0,"node_id":"n-05baa90780a27a2f","role":"sequence","name":"derivative of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by upward antidiagonals.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}]},{"id":"e-f0833858828f3c1d","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-daf15103e37c2e90","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":"63d1567f85136cab4e017374561ddc888672e645542f4caa13f88cb35ecf8a75","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-622af52a4165c278","role":"sequence","name":"shift_x_plus_one of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by upward antidiagonals.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-72b3e2064d6982f9","kind":"fingerprint_related","node_a_id":"n-daf15103e37c2e90","node_b_id":"n-d0bf5f74983ec343","fingerprint":"f6-e7a207c59c800cef","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e7a207c59c800cef"},"other_id":"n-d0bf5f74983ec343","other":{"id":"n-d0bf5f74983ec343","name":"mobius_x_over_x_plus_one of theta_plus_one of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.","first_index":1,"fingerprint":"f6-e7a207c59c800cef","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["0","2"],["0","2","5"]],"card_rows":[["1"],["0","2"],["0","2","5"],["0","0","6","10"]],"row_sums_preview":["1","2","7","16","41","94","219","492"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-392f49b496f79fbf","steps":["\u03b8+1","x \u21a6 x/(x+1)"],"edges":["e-cd00da36b6a89aa7","e-6611407bd9144a1d"],"text":"A026729 \u2192 \u03b8+1 \u2192 x \u21a6 x/(x+1)"},"headline":"A026729 \u2192 \u03b8+1 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-968c26fd321c6e73","kind":"fingerprint_related","node_a_id":"n-57c5a2449091b6f8","node_b_id":"n-daf15103e37c2e90","fingerprint":"f6-e7a207c59c800cef","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e7a207c59c800cef"},"other_id":"n-57c5a2449091b6f8","other":{"id":"n-57c5a2449091b6f8","name":"derivative of Triangle read by rows: T(n,k) is the number of ways to tile a 2 X n rectangle with k pieces of 2 X 2 tiles and n-2k pieces of 1 X 2 tiles (0 <= k <= floor(n/2)).","first_index":0,"fingerprint":"f6-e7a207c59c800cef","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],[],["1"]],"card_rows":[["1"],["2"],["5","2"],["10","6"]],"row_sums_preview":["0","0","1","2","7","16","41","94"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7d332fcb7bf9a0f7","steps":["D"],"edges":["e-44388154107108dc"],"text":"A128100 \u2192 D"},"headline":"A128100 \u2192 D","chain":null}},{"id":"l-e9bc52a3e82fb00b","kind":"fingerprint_related","node_a_id":"n-daf15103e37c2e90","node_b_id":"n-c99a4c5c96f874a0","fingerprint":"f6-e7a207c59c800cef","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e7a207c59c800cef"},"other_id":"n-c99a4c5c96f874a0","other":{"id":"n-c99a4c5c96f874a0","name":"shift_x_plus_one of derivative of Triangle read by rows: T(n,k) = binomial(n+1-k,k) (n >= 0, 0 <= k <= n).","first_index":0,"fingerprint":"f6-e7a207c59c800cef","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["1"],["2"]],"card_rows":[["1"],["2"],["5","2"],["10","6"]],"row_sums_preview":["0","1","2","7","16","41","94","219"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-569a801b809b6abd","steps":["D","x \u21a6 x+1"],"edges":["e-45a7ac4082b2eab9","e-11fd25a195e1820e"],"text":"A169803 \u2192 D \u2192 x \u21a6 x+1"},"headline":"A169803 \u2192 D \u2192 x \u21a6 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:   0;\n   2:   0;\n   3:   1;\n   4:   2;\n   5:   5,   2;\n   6:  10,   6;\n   7:  20,  18,   3;\n   8:  38,  44,  12;\n   9:  71, 102,  42,  4;\n  10: 130, 222, 120, 20;\n","recurrence_sentence":"The row polynomials satisfy (n-3)*P(n,x) = (3*n-10+(2*n-8)*x)*P(n-1,x) - (5+9*x+4*x^2)*d/dx P(n-1,x) for n >= 4, with P(1,x) = 0, P(2,x) = 0, P(3,x) = 1.","mathematica":"P[1, x_] := 0;\nP[2, x_] := 0;\nP[3, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((3*n - 10 + (2*n - 8)*x)*P[n - 1, x] - (5 + 9*x + 4*x^2)*D[P[n - 1, x], x])/(n - 3)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 0; v[3] = 1; for(n = 4, N, my(i = n); v[i] = ((3*n - 10 + (2*n - 8)*x)*v[i - 1] - (5 + 9*x + 4*x^2)*deriv(v[i - 1], x))/(n - 3)); 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-c8b3f39d03de798f","steps":["D","x \u21a6 x+1"],"edges":["e-076e29b3e57f3252","e-8b7b972ac21d1c05"],"text":"A052553 \u2192 D \u2192 x \u21a6 x+1"},"headline":"A052553 \u2192 D \u2192 x \u21a6 x+1","chain":null}