{"id":"n-8f1dfaae790c20d9","revision":0,"first_index":1,"fingerprint":"f6-3d8013fbbe82cb22","name":"shift_x_plus_one of shift_x_plus_one 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":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["1"],["3","1"],["5","2"],["11","7","1"],["21","16","3"],["43","41","12","1"],["85","94","34","4"],["171","219","99","18","1"],["341","492","261","60","5"],["683","1101","678","195","25","1"],["1365","2426","1692","576","95","6"],["2731","5311","4149","1644","340","33","1"],["5461","11528","9959","4488","1106","140","7"],["10923","24881","23568","11925","3430","546","42","1"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","1","4","7","19","40","97","217","508","1159","2683","6160","14209","32689","75316"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["-9","-4"],["5","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-18","-17","-4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":171565,"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-e52c03af06472321"]},{"id":171566,"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-e52c03af06472321"]},{"id":171567,"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-e52c03af06472321"]}],"region_exact":{},"incoming_edges":[{"id":"e-e52c03af06472321","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-8f1dfaae790c20d9","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":"5edbf45b8093fc64d7b9781167f2a073774f68edf40c260646c04b914e6fe856","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-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-7e2e2527b4ca2061","kind":"fingerprint_related","node_a_id":"n-ac04e52510a48bb5","node_b_id":"n-8f1dfaae790c20d9","fingerprint":"f6-3d8013fbbe82cb22","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-3d8013fbbe82cb22"},"other_id":"n-ac04e52510a48bb5","other":{"id":"n-ac04e52510a48bb5","name":"Triangular array read by rows: row n gives the coefficients of the polynomial p(n,x) defined in Comments.","first_index":0,"fingerprint":"f6-3d8013fbbe82cb22","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["3","1"],["5","2"]],"card_rows":[["1"],["3","1"],["5","2"],["11","7","1"]],"row_sums_preview":["1","4","7","19","40","97","217","508"],"regions":["(-inf, -2]","(-inf, -1]"],"region_exact":{},"oeis_ids":["A249139"],"derivation":{"root":"n-ac04e52510a48bb5","steps":[],"edges":[],"text":"A249139"},"headline":"A249139: Triangular array read by rows: row n gives the coefficients of the polynomial p(n,x) defined in Comments.","chain":null}},{"id":"l-adee6397528f02a5","kind":"fingerprint_related","node_a_id":"n-8f1dfaae790c20d9","node_b_id":"n-a25d5a5440f7c216","fingerprint":"f6-3d8013fbbe82cb22","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-3d8013fbbe82cb22"},"other_id":"n-a25d5a5440f7c216","other":{"id":"n-a25d5a5440f7c216","name":"mobius_x_over_x_plus_one of Another version of Fibonacci-Pascal triangle A037027.","first_index":0,"fingerprint":"f6-3d8013fbbe82cb22","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["0","1"],["0","1","3"]],"card_rows":[["1"],["0","1"],["0","1","3"],["0","0","2","5"]],"row_sums_preview":["1","1","4","7","19","40","97","217"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-e7c079f81f452bf3","steps":["x \u21a6 x/(x+1)"],"edges":["e-537bf9237ec77614"],"text":"A111006 \u2192 x \u21a6 x/(x+1)"},"headline":"A111006 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-f2026cfcda841e37","kind":"fingerprint_related","node_a_id":"n-fe5f0667f4ce56c9","node_b_id":"n-8f1dfaae790c20d9","fingerprint":"f6-3d8013fbbe82cb22","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-3d8013fbbe82cb22"},"other_id":"n-fe5f0667f4ce56c9","other":{"id":"n-fe5f0667f4ce56c9","name":"shift_x_plus_one 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":"f6-3d8013fbbe82cb22","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1"],["3","1"]],"card_rows":[["1"],["1"],["3","1"],["5","2"]],"row_sums_preview":["1","1","4","7","19","40","97","217"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7d332fcb7bf9a0f7","steps":["basis A007318"],"edges":["e-10b46a3f983f65c9"],"text":"A128100 \u2192 basis A007318"},"headline":"A128100 \u2192 basis A007318","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:   1;\n   3:   3,   1;\n   4:   5,   2;\n   5:  11,   7,   1;\n   6:  21,  16,   3;\n   7:  43,  41,  12,  1;\n   8:  85,  94,  34,  4;\n   9: 171, 219,  99, 18, 1;\n  10: 341, 492, 261, 60, 5;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = (5*n-9+(2*n-4)*x)*P(n-1,x) - (18+17*x+4*x^2)*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[((5*n - 9 + (2*n - 4)*x)*P[n - 1, x] - (18 + 17*x + 4*x^2)*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] = ((5*n - 9 + (2*n - 4)*x)*v[i - 1] - (18 + 17*x + 4*x^2)*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-c8b3f39d03de798f","steps":["x \u21a6 x+1","x \u21a6 x+1"],"edges":["e-13fa50edf65b735e","e-e52c03af06472321"],"text":"A052553 \u2192 x \u21a6 x+1 \u2192 x \u21a6 x+1"},"headline":"A052553 \u2192 x \u21a6 x+1 \u2192 x \u21a6 x+1","chain":null}
